Meaning And Necessity A Study In Semantics And Modal Logic

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MEANING AND NECESSITY

MEANING

AND NECESSITY

A Study in Semantics and Modal Logic

By RUDOLF CARNAP

/‘wjfinii ij PliiloiOpliy Hi the I'liiienit) uj Ckuai^v

I’HE UNIX'ERSITV Ol' (UIICLXCO PRESS
CHICAC.O ' ILLINOIS

The University of Chicago Press, Chicago 37
Cambridge University Press, London, N.W. 1, England
VV. J. Gage & Co., Limited, Toronto 2B, Canada

Copyriglil 1947 by the University of Chicago. AH rights reserved.
Published 1947. Second Impression 1948. Composed and printed by
The University of Chicago Press, Chicago, Illinois, USA.

PREFACE

The main purpose of this book is the development of a new method for
the semantical analysis of meaning, that is, a new method for analyzing
and describing the meanings of linguistic expressions. This method, called
the method of extension and intension, is developed by modifying and ex-
tending certain customary concepts, especially those of class and property.
The method will be contrasted with various other semantical methods
used in traditional philosophy or by contemporary authors. These other
methods havione characteristic in common : They all regard an expression
in a language as a name of a concrete or abstract entity. In contradistinc-
tion, the method here proposed takes an expression, not as naming any-
thing, but as possessing an intension and an extension.

This book may be regarded as a third volume of the series which I have
called “Studies in Semantics”, two volumes of which were published ear-
lier. However, the present book does not presuppose the knowledge of its
predecessors but is independent. The semantical terms used in the present
volume are fully explained in the text. The present method for defining the
L-terms (for example, ‘L-true’, meaning ‘logically true’, ‘analytic’) differs
from the methods discussed in the earlier Introduction to Semantics. I now
think that the method used in this volume is more satisfactory for lan-
guages of a relatively simple structure.

After meaning analysis, the second main topic discussed in this book is
modal logic, that is, the theory of modalities, such as necessity, contin-
gency, possibility, impossibility, etc. Various systems of modal logic have
been proposed by various authors. It seems to me, however, that it is not
possible to construct a satisfactory system before the meanings of the
modalities are sufiiciently clarified. I further believe that this clarification
can best be achieved by correlating each of the modal concepts with a cor-
responding semantical concept (for example, necessity with L-truth). It
will be seen that this method also leads to a clarification and elimination
of certain puzzles which logicians have encountered in connection with
modalities. In the Preface to the second volume of “ Studies in Semantics,”
I announced my intention to publish, as the next volume, a book on
modal logic containing, among other things, syntactical and semantical
systems which combine modalities with quantification. The present book,
however, is not as yet the complete fulfilment of that promise : it contains

VI

PREFACE

only analyses and discussions of modalities, preliminary to the construc-
tion of modal systems. The systems themselves are not given here. In an
article published elsewhere (see Bibliography), I have stated a calculus
and a semantical system combining modalities with quantification, and
have summarized some of the results concerning these systems. A more
comprehensive exhibition of results already found and those yet to be
found must be left for another time.

The investigations of modal logic which led to the methods developed
in this book were made in 1942, and the first version of this book was writ-
ten in 1943, during a leave of absence granted by the University of Chi-
cago and financed by the Rockefeller Foundation. To each of these insti-
tutions I wish to express my gratitude for their help. Professors Alonzo
Church and W. V. Quine read the first version and discussed it with me in
an extensive correspondence. I am very grateful to both for the stimula-
tion and clarification derived from this discussion, and to Quine also for a
statement of his view and, in particular, of his reaction to my method of
modal logic. This statement is quoted in full and discussed in detail in the
penultimate section of this book. I am also indebted to Professors Carl G.
Hempel and J. C. C. McKinsey for some helpful comments. To Miss
Gertrude Jaeger I am grateful for expert help in the preparation of the
manuscript.

CaicAGo
November 1946

R. C.

CONTENTS

I. The Method or Extension and Intension . . . . i

1. Preliminary Explanations . .... 2

2. L-Concepts 7

3. Equivalence and L-Equivalence 13

4. Classes and Properties 16

5. Extensions and Intensions ... 23

6. Extensions and Intensions of Sentences .... • 2.1

7. Individual Descriptions ... . . 32

8. Frege’s Method for Descriptions - 3,1

g. Extensions and Intensions of Individual Expressions . . 3 q

10. Variables ... 42

11. Extensional and Intensionai Contexts .... .46

12. The Principles of Interchangeability 51

13. Sentences about Beliefs . ■ • ■ ■ ■ S 3

14. Intensionai Structure . . . .... 56

IS- Applications of the Concept of Intensionai Structure 50

16. Lewis’ Method of Meaning Analysis ... 64

II. L-Determinacy .... . . . 6g

17. L-Determinate Designators . ... 60

18. The Problem of L-Detcrminacy of Individual Exprc.ssions 73

ig. Definition of L-Determinac\' of Individual Exj)ressions ... 78

20. L-Determinacy of Predicators 8i

21. Logical and Descriptive Signs 83

22. L-Detcrminatc Intensions ... ,88

23. Reduction of Extensions to Intensions go

III. 'I'he Method of the Name-Relation , , gb

24. The Name-Relation g6

25. An Ambiguity in the Method of the Name-Relation 100

26. The Unnecessar}' Duplication of Names ... 106

27. Names of Classes .... in

28. Frege's Distinction between Nominatum and Sense 118

2g. Nominatum and Sense: Extension and Intension 124

30. The Disadvantages of Frege’s Method .... 129

31. The Antinomy of the Name- Relation .... 133

32. Solutions of the Antinomy .... 136

vii

CONTENTS

viii

IV. On Metalanguages fok Semantics X4S

33. The Problem of a Reduction of the Entities 14S

34. The Neutral Metalanguage M' iS3

35. M' Is Not Poorer than M 157

36. Neutral Variables in M' 160

37. On the Formulation of Semantics in the Neutral Metalanguage M' 161

38. On the Possibility «f an Extensional Metalanguage for Semantics 168

V. On THE Logic or Modalities 173

39. Logical Modalities 173

40. Modalities and Variables .... 177

41. Semantical Rules for the Modal Sj stem Sa 182

42. Modalities in the Word Language 186

43. Modalities and V'ariables in the Word Language igi

44. Quine on Modalities 193

45. Conclusions 202

Bibuography ... 203

Index 207

CHAPTER I

THE METHOD OF EXTENSION AND INTENSION

A method of semantical meaning analysis is developed in this chapter. It is
applied to those expressions of a semantical system 5 which we call designators;
they include (declarative) sentences, individual expressions (i.e., individual
constants or individual descriptions) and predicators (i.e., predicate constants
or compound predicate expressions, including abstraction expressions) , We start
with the semantical concepts of truth and L-truth (logical truth) of sentences
(§§ I, 2). It is seen from the definition of L-truth that it holds for a sentence if
its truth follows from the semantical rules alone without reference to (extra-
linguistic) facts (§ 2). Two sentences are called (materially) equivalent if both
are true or both are not true. The use of this concept of equivalence is then
extended to designators other than sentences. Two individual expressions are
equivalent if they stand for the same individual. Two predicators (of degree
one) are equivalent if they hold for the same individuals. L-equivalence (logical
equivalence) is defined both for sentences and for other designators in such a
manner that it holds for two designators if and only if their equivalence follows
from the semantical rules alone. The concepts of equivalence and L-equivalence
in their extended use are fundamental for our method (§ 3).

If two designators are equivalent, we say also that they have the same
extension. If they are, moreover, L-equivalent, w'e say that they have also the
same intension (§ 5). Then we look around for entities which might be taken
as extensions or as intensions for the various kinds of designators. We find that
the following choices are in accord with the two identity conditions just stated.
We take as the exrension of a predicator the class of those individuals to which
it applies and, as its intension, the property which it expresses; this is in ac-
cord with customary conceptions (§ 4). As the exte n-sinn of f t jfpntpnre we take
its truth-value (truth or falsity); as its intension, the proposition expressed by
it (§6). Finally, the ex tension of an individual expression is the individual to
which it refers; its infension is a concept of a new kind expressed by it, which we
call an individual concept (§§ 7-9). These conceptions of extensions and inten-
sions are justified by their fruitfulness; further definitions and theorems apply
equally to extensions of all types or to intensions of all types.

A sentence is said to be extensional with respect to a designator occurring in
it if thTextension ot the sentence is a t urml dn'oTtlte eiteQsion of the designa-
tor. tha t is to say, if the re olaceme nFot the designator by an equivalent one
transfor ms the whole sentence into an e q uivalent one. A senteloce F tsid toTir
intensional with r espect to a jjpgi^natnr nrruj^ririg in 'it if it U not extensional
~ahd i f its intension is a function of th e intension oTthg designator, tTialjs to
s ay, if the replacement of this designator bv an L-eqii ivaTent one transforms the
whole sente nce in to an L-eauivalent one. A modal se n tence (for examnle. ‘it is
necessary that -is- intensinnal with respecTto its_subsentence (§ ri). A
psychological sentence like ‘JoJiq h a l iewiahat j f j ^aining now’ is neither ex-
fEnsltmal-nor-ifrtenslSnaT’v^^ respect to its subsentence~(| ij). The proBlCTi of
the semantical aflalysis of these- Mief-setilences^is solved with the help of the
com

2

I. THE METHOD OF EXTENSION AND INTENSION

§ 1. Preliminary Explanations

This section contains explanations of a symbolic language system Si, which
will later serve as an object language for the illustrative application of the
semantical methods to be discussed in this book. Further, some semantical con-
cepts are explained for later use; they belong to the semantical metalanguage
M, which is a part of English. Among them are the concepts of truth, falsity, and
(material) equivalence, applied to sentences. The term ‘des ignator’ is -uttroduced
for a ll those exp ressions to which a semantical m eaning analysis is applied, the
term .wiU. Jje4ise3'li£i.e especially for sentences, predirators (i.eT) predicate "ex-
pressions.), and individuaLexpresSonS. ■

The chief task of this book will be to find a suitable method for the
semantical analysis of meaning, that is, to find concepts suitable as tools
for this analysis. The concepts of the intension and the extension of an ex-
pression in language will be proposed for this purpose. They are .anal-
ogous to the cust omary concepts of property and class b ut will be ap-
plied in a more general way to various types of expressions, including
sentences and individual expressions. The two concepts will be explained
and discussed in chapters i and ii.

The customary concept of name-relation and the distinction sometimes
made since Frege between the entity named by an expression and the
sense of the expression will be discussed in detail in chapter iii. The pair
of concepts, extension-intension, is in some respects similar to the pair of
Frege’s concepts; but it will be shown that the latter pair has serious dis-
advantages which the former avoids. The chief disadvantage of the meth-
od applying the latter pair is that, in order to speak about, say, a property
and the corresponding class, two different expressions are used. The meth-
od of extension and intension needs only one expression to speak about
both the property and the class and, generally, one expression only to
^ speak about an intension and the corresponding extension.

In chapter iv, a metalanguage will be constructed which is neutral with
regard to extension and intension, in the sense that it speaks not about a
property and the corresponding class as two entities but, instead, about
one entity only; and analogously, in general, for any pair of an intension
and the corresponding extension. The possibility of this neutral language
shows that our distinction between extension and intension does not pre-
suppose a duplication of entities.

In chapter v, some questions concerning modal logic are discussed on
the basis of the method of extension and intension.

My interest was first directed toward the problems here discussed when
I was working on systems of modal logic and found it necessary to clarify
the concepts which will be discussed here under the terms of ‘ extension’

§ 1. PRELIMINARY EXPLANATIONS

3

and ‘intension’ and related concepts which have to do with what is usual-
ly called the values of a variable. Further stimulation came from some
recent publications by Quine' and Church,' whose discussions are valu-
able contributions to a clarification of the concepts of naming and
meaning.

Before we start the discussion of the problems indicated, some explana-
tions will be given in this section concerning the object languages and the
metalanguage to be used. We shall take as object languages mostly sym-
bolic languages, chiefly three semantical language systems, S„ Sj, and Sj,
and occasionally also the English word language. For the sake of brevity,
not all the rules of these symbolic systems will be given, but only those of
their features will be described which are relevant to our discussion. S,
will now be described; S, is an extension of it that will be explained later
(§ 41) ; Sj will be described in § 18.

The system S, contains the customary connectives of negation
(‘not’), disjunction' ‘V’ (‘or’), conjunction (‘and’), conditional (or
material implication) ‘ D ’ (‘if . . . then . . .’), and biconditional (or ma-
terial equivalence ) ' = ' (‘if and only if’). The only variables occurring are
individual variables ‘x’, ‘y’, ‘z’, etc. For these variables the customary
universal and existential quantifiers are used: ‘(a:)(. . x . .)’ (‘for every
X, ..*..’) and ‘( 3 ar)(, (‘there is an x such that . . x . .’). All

sentences in Si and the other systems are closed (that is, they do not con-
tain free variables). In addition to the two quantifiers, two other kinds of
operators occur: the iota-operator for individual descriptions (‘(1*)
(. . X . .)’, ‘the one individual x such that . . x . and the lambda-
operator for abstraction expressions (‘(Xx)(. . x . .)’, ‘the property (or
class) of those x which are such that . . x . .’) . If a sentence consists of an
abstraction expression followed by an individual constant, it says that the
individual has the property in question. Therefore, ‘(Xx)(. . x . .)a’
means the same as ‘ . . a . .’, that is, the sentence formed from ‘. . x . .’
by substituting ‘a’ for ‘x’. The rules of our system will permit the trans-
formation of ‘(Xx)(. . X . .)a’ into ‘. . a . .’ and vice versa; these trans-
formations are called conversions.

Si contains descriptive constants (that is, nonlogical constants) of indi-

■ [Notes] (see Bibliography at the end of this book). Quine’s views concerning the name-
relation (designation) will be discussed in chap, iiij and the conclusions which he draws from
them for the problem of quantification in modal sentences will be discussed in chap. v.

‘ [Review C.] and [Review Q.j. Church’s conceptions will be discussed in chap, iii, in con-
nection with those of Frege. Church’s contributions are more important than is indicated by
the form of their publication as reviews. It is to be hoped that he will soon find the opportu-
nity for presenticig his conception in a more comprehensive and systematic form.

4

I. THE METHOD OF EXTENSION AND INTENSION

vidual and predicate types. The number of predicates in S, is supposed to
be finite, that of individual constants may be infinite. For some of these
constants, which we shall use in examples, we state here their meanings
by semantical rules which translate them into English.

1 - 1 . Rules of designation for individual constants
‘s’ is a symbolic translation of ‘Walter Scott’,

‘ w’ — ‘ (the book) Waverley’.

1 - 2 . Rules of designation for predicates
‘Hac’ — ‘x is human (a human being)’,

‘RAx ’ — 'x is a rational animal’,

‘Fa;’ — ‘x is (naturally) featherless’,

‘Bx’ — ‘x is a biped’,

‘Axy’ — ‘x is an author of y’.

The English words here used are supposed to be understood in such a way
that ‘human being’ and ‘rational animal’ mean the same. Further, we
shall use ‘a’, ‘b’, ‘c’, as individual constants, and ‘P’, ‘Q’, as predicator
constants (of level one and degree one) ; the interpretation of these signs
will be specified in each case, or left unspecified if not relevant for the
discussion.

In order to speak about any object language — here the symbolic language
systems Si, etc. — we need a metalanguage. We shall use as our metalan-
guage M a suitable part of the English language which contains transla-
tions of the sentences and other expressions of our object languages (for
example, the translations stated in i-i and 1 - 2 ), names (descriptions) of
those expressions, and special semantical terms. For the sake of simplicity,
we shall usually construct a name of an expression in the customary way
by including it in single quotation marks. In order to speak about expres-
sions in a general way, we often use Ԥ1/, etc., for expressions of any
kind and‘@j’, ‘@/, etc., for sentences, sometimes also blanks like
‘- -’, etc., and blanks with a variable, e.g., ‘. . x . for an expression in
which that variable occurs freely. If a German letter occurs in an expres-
sion together with symbols of the object language, then the latter ones are
used autonymously, i.e., as names for themselves.^ Thus, we may write in
M, for instance, ‘31, s 21/; this is meant as referring to that expression
of the object language which consists of the expression 21, (whatever this
may be, e.g., ‘Hs’) followed by the sign ‘ = followed by the expression
21,. (In symbolic formulas both in the object languages and in M, paren-
theses will often be omitted under the customary conditions.) The term

> See (Syntax), § 43.

§ 1. PRELIMINARY EXPLANATIONS

S

* sentence' will be used in the sense of ‘declarative sentence’. The term
‘sentential matrix’ or, for short, 'matrix' will be used for expressions
which are either sentences or formed from sentences by replacing indi;
vidual constants with variables. (If a matrix contains any number of free
occurrences of n different variables, it is said to be of degree n; for ex-
ample, ‘ hxy V P*’ is of degree twn; the sentpnres are the, matrices of de-
gree zero) . A sentence consisting of a predicate of degree n followed by n
individual constants is called an atomic sentence (e.g., ‘Pa’, ‘Abe’).

A complete construction of the semantical system S„ which cannot be
given here, would consist in laying down the following kinds of rules;

(1) rules of formation, determining the admitted forms of sentences;

(2) rules of designation for the descriptive constants (e.g. i-i and 1-2);

(3) rules of truth, which we shall explain now; (4) rules of ranges, to be ex-
plained in the next section. Of the rules of truth we shall give here only
three examples, for atomic sentences (t-3), for'V’ (1-5), and for' =’ (1-6).

1-3. Rule of truth for the simplest atomic sentences. An atomic sentence
in S, consisting of a predicate followed by an individual constant is true
if and only if the individual to which the individual constant refers posses-
ses the property to which the predicate refers.

This rule presupposes the rules of designation. It yields, together with
rules i-i and 1-2, the following result as an example;

1-4. The sentence ‘Bs’ is true if and only if Scott is a biped.

1-6. Rule of truth for 'V'. A sentence V is true in S, if and only if at
least one of the two components is true.

1-6. Rule of truth for ‘ = ’. A sentence ©, = ©/ is true if and only if either
both components are true or both are not true.

There are some further rules of truth for the other connectives, cor-
responding to their truth-tables, and for the quantifiers; another example
of a rule of truth will be given in 3-3. The rules of truth together constitute
a recursive definition fo r ' true in S,’ . becau se they determine, in combina-
Uon with the r ules of design adopj for every sentence in S, a sufScient and
n ecessa^ conji^n of its truth (a s is given for 'JBsiin-r^al . Thereby they
give an interpretat io n for every sentegee. Thus, for example, we learn from
the rules that the sentence ‘Bs’ says that (in other words, expresses the
proposition that) Scott is a biped. For the purposes of our discussion it is
not necessary to give the whole definition of truth. ■' It will sufhee to pre-

* The first definition of the semantical concept of truth was given by Tarski [Wahrheitsbe-
griff]; I have given a slightly different form in [I|, § 7, For nontechnical discussions of the na-
ture of the semantical concept of truth see Tarski (Truth] and my [Remarks].

6

I. THE METHOD OF EXTENSION AND INTENSION

suppose that the term ‘true’ is defined in such a manner that it has its
customary meaning as applied to sentences. More specifically, we presup-
pose that a statement in M saying that a certain sentence in S, is true
means the same as the translation of this sentence;® for example, ‘the
sentence ‘Hs’ is true in Si’ means the same as ‘Walter Scott is human’.
On the basis of ‘true’, some further semantical terms are defined as fol-
lows, with respect to any semantical system 5, e.g., S„ etc.

1-7. Definition. is false (in S) =Df is true (in S).

Thus ‘false’ has here its ordinary meaning.

1-8. Definition. is equivalent to ©y (in S) =Df ®. = is true (in S).

This definition, together with the rule of truth for‘ = ’ (i-6), yields this
result:

1-9. Two sentences are equivalent if and only if both have the same truth-
value, that is to say, both are true or both are false.

It is to be noticed that the term ‘equivalent’ is here defined in such a
manner that it means merely agreement with respect to truth- value (truth
or falsity), a relation which is sometimes called ‘material equivalence’.
The term is here not used, as in ordinary language, in the sense of agree-
ment in meaning, sometimes called ‘logical equivalence’; for the latter
concept we shall later introduce the term ‘L-equivalent’ (2-3C).

I propose to use the term 'designator* for all those expressions to
which a semantical analysis of meaning is applied, the class of designators
thus being narrower or wider according to the method of analysis used.
[The word ‘ mea ning’ is h ere always understood in the sense of ‘designa-
tive mea nin g’ snmp ti m i » R nlfo rnllr- d _‘ cognitive’, ‘theoretical’, ‘referen-
tial’, or ‘infor mative’ a.s di^itingniji hed from other meaning components ,
e. g.. emotive nr motivative meaning. Thus here we have to do only with
dec larative sentences and their parts .] Our method takes as designators at
le^t sentences, predicators* (i.e., predicate expressions, in a wide sense,

s For detailed discussions of this characteristic of the semantical concept of truth, see
Tarski [Truth] and my [Remarks], § 3.

‘ Some terms with the ending '-tor’ for kinds of expressions are customary, e.g., ‘functor’,
‘operator’. The terms ‘predicator’ and ‘designator’ are formed in analogy to them. A still
wider use of the same ending might be taken into consideration with the aim of making the
terminology in the metalanguage somewhat more uniform. For this book, only the two terms
mentioned are adopted; but the following terms would seem to me quite suitable, too:
‘descriptor’ (for the customary ‘description’), ‘abstractor’ (for ‘abstraction expression’),
‘connector’ (for ‘connective’). Other terms might seem more questionable, but perhaps stiil
worth consideration, e.g., ‘individuator’ (for ‘individual expression’), ‘propositor’ or ‘stator’
(for ‘(declarative) sentence’), ‘conceptor’ (for ‘concept expression,’ i.e., ‘designator other than
sentence’). Morris, [Signs], uses a number of terms with ‘-tor’ (or ‘-or’), among them some of
those mentioned here, for kinds of expressions or, more generally, of signs including non-
linguistic signs.

§2. L-CONCEPTS

7

including class expressions), functors (i.e., expressions for functions in
the narrower sense, excluding propositional functions), and individual ex-
pressions; other types may be included if desired (e.g., connectives, both
extensional and modal ones) . The term ‘designator’ is not meant to imply
that these expressions are names of some entities (the name-relation will
be discussed in S 2 a). hut merely that they have, so to speak, an inde-
pendent meaning, at least independent to some degree. Only (declarative)
sSitenc es have a (designative) meaning in the strictest sense, a meaning of
the highest Hpgrpp nf inde pendence All Other expression s derive what-
meaning they hav^om the wav in which they contribute to the meaning
of t he sentencp-; in which they occur. O ne might perhaps distinguish —
m a vague way — different degrees of independence of this derivative
meaning. Thus, for instance, I should attribute a very low degree to
‘ (’, somewhat more independence to' V’, still more to ‘ (in an arithmet-

ical language), still more to'H’ ('human’) and ‘s’ (‘Scott’); I should not
know which of the last two to rank higher. This order of rank is, of course,
highly subjective. And where to make the cut between expressions^th no _
or little in dependence of meaning (‘ svncategorematic’ in traditional
terminol ogy) and tho s e with a high degree of independence, to be taken a ^
~designat ors. seems more oHess a m atter of convention. If a metalanguage
i^Hprjfjprl iipniiythenlitjeg ms con venierit^ t o take as o^ignafors at least
the expressions of. aJl those type s, but not necessarily only thn.se^far which
jther e are vari ables in the metalanguage_(compare [I|. § 12, and refe renc es
to Quine, below^ at the beginning of § 10).

§ 2 . L-Concepts

By the explication of a familiar but vague concept we mean its replacement
by a new exact concept; the former is called explicandum, the latter explicatum.
The co ncep t of L-trulh is here defined as an exulicatum for what philosophers
call logical or necessary or analytic truth. The definition leads to the result that
a sentence in a semantical system is L-true if- and only if the semantical rules
of the system suffice for establishing its truth. The concepts of L-falsity, L-
implication, and L-eguivalence are defined as explicata for logical falsity, logical
implication or entailmcnt, and mutual logical implication, respectively. A sen-
tence is called L-d eterminate i f it is either L-true or L-false; otherwise it is called
L-indeterminffle or factual. The latter concept is an explicatum for vyhat Kapt
called synthetic judgments.' A sentence is cmled Z’-lrSe TriFIi^ true but not L-
TrueTT-trutn is an explicatum for what is known as factual or synthetic or
contingent truth. The concepts of F-falsity, F-implication, and F-equivalence
are defined analogously.

The task of making more exact a vague or not quite exact concept used
in everyday life or in an earlier stage of scientific or logical development.

8 I. THE METHOD OF EXTENSION AND INTENSION

or rather of replacing it by a newly constructed, more exact concept, be-
longs among the most important tasks of logical analysis and logical con-
struction. We call this the task of explicating, or of giving an explication
for, the earlier concept; this earlier concept, or sometimes the term used
for it, is called the explicandum; and the new concept, or its term,
is called an explicatum of the old one.' Thus, for instance, Frege and,
later, Russell took as explicandum the term ‘ two’ in the not quite exact
meaning in which it is used in everyday life and in applied mathematics;
they proposed as an explicatum for it an exactly defined concept, namely,
the class of pair-classes (see below the remark on (i) in § 27) ; other
logicians have proposed other explicata for the same explicandum. Many
concepts now defined in semantics are meant as explicata for concepts
earlier used in everyday language or in logic. For instance, the semantical
concept of truth has as its explicandum the concept of truth as used in
everyday language (if applied to declarative sentences) and in all of
traditional and modern logic. Further, the various interpretations of de-
scriptions by Frege, Russell, and others, which will be discussed in §§ 7
and 8, may be regarded as so many different explications for phrases of
the form ‘ the so-and-so’ ; each of these explications consists in laying down
rules for the use of corresponding expressions in language systems to be
constructed. The interpretation which we shall adopt following a sug-
gestion by Frege ( §8, Method Illb) deviates deliberately from the mean-
ing of descriptions in the ordinary language. Generally speaking, it is not
required that an explicatum have, as nearly as possible, the same meaning
as the explicandum ; it should, however, correspond to the explicandum in
such a way that it can be used instead of the latter.

The L-terms (‘L-true’, etc.) which we shall now introduce are likewise
intended as explicata for customary, but not quite exact, concepts.
^L-true' is meant as an explicatum for what Leibniz called necessary
truth and Kant analytic truth. We shall indicate here briefly how this
and the other L-terms can be defined. In the further discussions of this
book, however, we shall not make use of the technical details of the fol-
lowing definitions but only of the fact that ‘L-true’ is defined in such a
way that the requirement stated in the subsequent convention 2-1 is ful-
filled. This is in accord with the purpose of this book, which is intended
not so much to carry out exact analyses of exactly constructed systems
as to state informally some considerations aimed at the discovery of
concepts and methods suitable for semantical analysis.

1 What is meant here by ‘explicandum’ and ‘explicatum’ seems similar to what Langford
means by ‘analysandum’ and ‘analysans’; see below, n. 42, p. 63.

§ 2 . L-CONCEPTS

9

We shall introduce the L-concepts with the help of the concepts of
state-description and range. Some ideas of Wittgenstein* were the start-
ing-point for the development of this method.’

A class of sentences in S, which c ontains for every atomic sentence
ei ther this sentence or its negation, hut not both, a nd no other sentences^
is called a state-descrittion in S. , because it obviously gives a complete
description of a possible state of the universe of individuals with respect
to all properties and relations expressed by predicates of the system. Thus
the state-descriptions represent Leibniz’ possible worlds or Wittgenstein’s
possible states of affairs.

It is easily possible to lay down semantical rules which determine for
every sentence in S, whether or not it holds in a given state-description.
That a sentence holds in a state-description means, in nontechnical terms,
that it would be true if the state-description (that is, all sentences belong-
ing to it) were true. A few examples will suffice to show the nature of these
rules: (i) an atomic sentence holds in a given state-description if and only
if i t belongs to it; ( 2) ~ ©, holds in a given state-description if and only
If does not hold in it; (3) ©, V ©,• holds in a state-description if and
only if either ©, holds in it or ©,• or both; (4) ©. = ©,■ holds in a state-
description if and only if either both ©,■ and ©,■ or neither of them hold
in it; (5) a universal sentence (e.g., ‘(x)(Pz)’) holds in a state-descrip-
tion if and only if all substitution instances of its scope (‘Pa’, ‘Pb’, ‘Pc’,
etc.) hold in it. Iota-operators and lambda-operators can be eliminated
(for the former, this will be shown later, see 8-2; for the latter, see the
explanation of conversion in § 1). Therefore, it is sufficient to lay down a
rule to the effect that any sentence containing an operator of one of these
kinds holds in the same state-descriptions as the sentence resulting from
the elimination of the operator.

The class of all those state-dpsrrip tions ir| which a given sen t ence ©,
holds i s c alled the range of ©j . All the rules together, of which we have just
given five examples, determine the range of any sentence in S, ; therefore,
they are called rules of ranges. By determining the ranges, they give, to-
gether with the rules of designation for the predicates and the individual
constants (e.g., i-i and 1-2) . an interl>retation for a .11 sentences in S„ since

* [TractatusJ; see also [I], p. 107.

’ The method which I shall use here is similar to, but simpler than, the one I have de-
scribed in [I], § ig, as procedure E. The simpler form is possible here because Si contains
atomic sentences for all atomic propositions. The procedure to be used here seems to me the
most convenient among those known at present for the semantical construction of a system
of deductive logic; I have used it, furthermore, for modal logic in [Modalitiesl and for induc-
tive logic, that is, the theory of logical probability or degree of confirmation in [Inductive].

lo

I, THE METHOD OF EXTENSION AND INTENSION

to know the meaning o f a. spnte nce is to know in which of the pos sible
c ases it would be true and in whic h not, as Wittgenstein has pointed
out.

The connection between these concepts and that of truth is as follows:
There is one and only one state-description which describes the actual
state of the universe ; it is that which contains all true atomic sentences
and the negations of those which are false. Hence it contains only true
sentences; therefore, we call it the true state-description. A sentence of
any form is true if and only if it holds in the true state-description. These
are only incidental remarks for explanatory purposes; the definition of
L-truth will not make use of the concept of truth.

The L-concepts now to be defined are meant as explicata for certain
concepts which have long been used by philosophers without being defined
in a satisfactory way. Our concept of L-truth is, as mentioned above, in-
tended as an explicatum for the familiar but vague concept of logical or
necessary or analytic truth as explicandum. This explicandum has some-
times been characterized as truth based on purely logical reasons, on
meaning alone, independent of the contingency of facts. Now the mean-
ing of a sentence, its interpretation, is determined bv the semantical rules
(t he rules of designation and the rules of ranges in the metho d explaine d
above). Therefore, it seems well in accord with the traSHonalconcept
which we take as explicandum, if we require of any explicatum that it
fulfil the following condition:

2-1. Convention. A sentence ©, is L-true in a semantical system A if
and only if is true in 5 in such a way that its truth can be established
on the basis of the semantical rules of the system S alone, without any
reference to (extra-linguistic) facts.

This is not yet a definition of L-truth, It is an informal formulation of
a condition which any proposed definition of L-truth must fulfil in order
to be adequate as an explication for our explicandum. Thus this conven-
tion has merely an explanatory and heuristic function.

How shall we define L-truth so as to fulfil the requirement 2 - 1 ? A way
is suggested by Leibniz’ conception that a necessary truth must hold in all
possible worlds. Since our state-descriptions represent the possible worlds,
this means that a sentence is logically true if it holds in all state-descrip-
tions. I'his leads to the following definition :

2-2. Definition. A sentence ®, is L-true (in S,) =Df holds in every
state-description (in Si).

I 2. L-CONCEPTS

if

The following consideration shows that the concept of L-truth thus
defined is in accord with the convention 2-1 and hence is an adequate
explicatum for logical truth. If ©,• holds in every state-description, then
the semantical rules of ranges suffice for establishing this result. [For ex-
ample, we see from the rules of ranges mentioned above that ‘ Pa’ holds
in certain state-descriptions, that ‘~Pa’ holds in all the other state-
descriptions, and that therefore the disjunction ‘PaV~Pa’ holds in
every state-description.] Therefore, the semantical rules establish also the
truth of <Bi because, if ©. holds in every state-description, then it holds
also in the true state-description and hence is itself true. If, on the other
hand, does not hold in every state-description, then there is at least
one state-description in which 0, does not hold. If this state-description
were the true one, 0, would be false. Whether this state-description is
true or not depends upon the facts of the universe. Therefore, in this
case, even if 0, is true, it is not possible to establish its truth without
reference to facts.

L-falsily is meant as an explicatum for logical or necessary falsity or
self-contradiction. L-implication is meant as explicatum for logical impli-
cation or entailment. L-equivaknce is intended as explicatum for mutual
logical implication or entailment. The definitions are as follows:

2-3. Definitions

a. 0, is L-false in (S,) =Df is L-true.

b. 0, L-implies 0j (in S,) =Df the sentence 0. D 0, is L-true.

c. 0. is L-equivalent to 0,- (in S.) =Df the sentence 0, = 0y is
L-true.

d. 0. is L-determinate (in S,) =Df is either L-true or L-false.

The following results follow easily from these definitions, together with
2-2:

2-4. 0. is L-false if and only if ©, does not hold in any state-description.
2-6. 0, L-implies 0,- if and only if 0, holds in every state-description
in which ©, holds.

2-6. 0, is L-equivalent to 0,- if and only if 0, and 0, hold in the same
state-descriptions.

The condition for L-falsity stated in 2-4 means, in effect, that 0< can-
not possibly be true. The condition for L-implication in 2-5 means that
it is not possible for @. to be true and for 0j to be false. The condition
for L-equivalence in 2-6 means that it is impossible for one of the two
sentences to be true and the other false. Thus these results show that
L-falsity, L-implication, and L-equivalence as defined by 2-33, b, c, may

12 I. THE METHOD OF EXTENSION AND INTENSION

indeed be regarded as adequate explicata for the explicanda mentioned
earlier.

We have seen that our concept of L-truth fulfils our earlier conven-
tion 2-1. Therefore, according to the definition 2-3d, a sentence is L-de-
terminate if and only if the semantical rules, independently of facts, suf-
fice for establishing its truth-value, that is, either its truth or its falsity.
This suggests the following definition, 2-7, as an explication for what Kant
called synthetic judgments. The subsequent result, 2-8, which follows
from the definition, shows that the concept defined is indeed adequate as
an explicatum.

2-7. Definition. is L-indeterminate or factual (in S,) =Df is not
L-determinate.

2-8. A sentence is factual if and only if there is at least one state-descrip-
tion in which it holds and at least one in which it does not hold.

The concept of F-truth to be defined by 2-9a is meant as an explicatum
for what is usually called factual or synthetic or contingent truth in con-
tradistinction to logical or necessary truth. The concepts defined by
2-9b, c, d, are meant as expUcata in an analogous way. The adequacy of
these F-concepts as explicata follows from the adequacy of the L-concepts.

2-9. Definitions

a. is F-true (in S,) =Df is true but not L-true.

b. ®i is F -false (in S.) is F-true.

c. F-implies (in S,) =Df D is F-true.

d. ©, is F-eguivalent to (in S,) =0/ ©, = is F-true.

The following are simple consequences of these and the earlier defini-
tions:

2-10. ©, is F-false if and only if ©, is false but not I.-false.

2-11. is F-equivalent to ©,■ if and only if is equivalent but not
L-equivalent to

As an example of F-truth, consider the sentence ‘Bs’. We found earlier
with the help of a rule of truth and rules of designation, that ‘ Bs’ is true
if and only if Scott is a biped (1-4). This result does not tell us whether
‘Bs’ is true or not; it merely states a sufficient and necessary condition
for the truth of the sentence ‘ Bs’. This is all we can learn about ‘ Bs’ from
the semantical rules alone. If we want to determine the truth-value of
‘Bs’, we have to go beyond the mere semantical analysis to the observa-
tion of facts. We see from 1-4 which facts are relevant: we must look at
the thing Walter Scott and see whether it is a biped. Observation shows

§ 3 . EQUIVALENCE AND L-EQUIVALENCE 13

that this is the case. Therefore, ‘ Bs’ is true. Since the semantical rules do
not suffice for establishing its truth, it is not L-true; hence it is F-true.

§ 3. Equivalence and L-Equivalence

The symbol ‘ = ’, customarily used between sentences, is used here also
between designators of other kinds, especially between predicators and between
individual expressions. ‘P s Q’ is to mean the same as ‘(i) (P* = Q*)'. ‘a = b'
is used, instead of the customary ‘a = b’, as an identity sentence, saying that
a is the same individual as b. Then the concepts of equivalence and L-equiva-
lence, previously applied to sentences only, are defined for designators of any
kind; these two concepts are fundamental in our method. Two designators are
said to be equivalent if the s -sentence connecting them is true; they are said
to be L~equivalenl if this sentence is L-true. It follows that ‘P' and ‘Q’ are
equivalent if they hold for the same individuals. And ‘a’ and ‘b’ are equivalent
if a is the same individual as b.

We have defined the terms ‘equivalent’ and ‘L-equivalent’ so far only
for sentences (1-8 and 2-3C). Now we shall extend their use so as to make
them applicable to all kinds of designators, especially also to predicators
and individual expressions. Extended in this way, the two concepts will
become the fundamental concepts in the method of semantical analysis
to be proposed here.

We begin by extending the use of the symbol ‘ s’. It is customary as a
connective between sentences. We shall use it in our systems between
two designators of any kind, but only if both designators are of the same
type. This use is introduced by the following rules of abbreviation. If the
extended use of ‘ s’ is taken as primitive, then suitable rules of ranges are
to be laid do^n which lead to the same results (for example, the result
that ‘P = Q’ has the same range as, and hence is L-equivalent to,
‘(a;)(Px s Qx)’). The reasons for choosing just these interpretations for
‘ s ’ with the various kinds of designators will soon become apparent.

The first rule introduces ‘ = ’ between predicators ;

3-1. Abbreviation

a. Let 21, and 21, be two predicators of the same degree n in S,.

21. = 21, for (Xx)(x,) . . (xJ[21vr,Xx . . x„ = 2ljX.Xx . . xj.

b. Hence for degree one:

2li = 21,- for (x)[2liX = 2[jx].

We shall use in S, the connective ‘ •’ also between predicators, but, for
the sake of a convenient notation, in a way different from the use of ‘ s ’
just introduced. The resulting expression (e.g., ‘P«Q’) is here taken as a
predicator, not as a sentence as in the case of ‘ = ’ (e.g., ‘P s Q’). We de-
fine it for degree one:

14

I. THE METHOD OF EXTENSION AND INTENSION

3-2. Abbreviation. Let 21, and 2Iy be two predicators of degree one in
S.. 2l..2lyfor (X*)[2l<a;,2Ija:].

Thus, for example, ‘F»B’ is short for ‘(Xa:)[Fa;*Ba:]’, and hence is an ex-
pression for the property of being a featherless biped.

Furthermore, we introduce ‘ ’ as a primitive sign of identity of indi-

viduals instead of the customary ‘ For this purpose we lay down the
following rule:

3-3. Jiule of truth. If 21, is an individual expression in S, for the individual
X and 21, • for y, then 21, = 21; is true if and only if x is the same individual
as y.

[If S is an extensional system containing, in distinction to S,, a predi-
cator variable ‘/’, then we can achieve the same result as 3-3 by defining
2l< = 21 j, in a way similar to Russell’s, as short for (/)[/(2Ii) = /(2ly)].]

If a system 5 contains, in distinction to S,, functors also, then ‘ = ’ can
be defined for them in a way similar to the above definition for predica-
tors. The method may be indicated briefly by stating the definition for the
simplest type, namely, functors for singulaiy functions from individuals
to individuals; the definitions for other types are analogous. This defini-
tion will not be used in our further discussions.

3-4. Abbreviation. For functors 2l< and 21,- in S:

21,' = 21,- for (a:)[2La; =

[Note that here on the right-hand side the sign ‘ s ’ stands, not between
sentential matrices, as in 3-ib, but between full expressions of functors,
which are for this type individual expressions.]

Now we shall define ‘equivalent’, ‘L-equivalent’, and ‘ F-equivalent’ in
a general way for all kinds of designators.

3-6. Definitions. Let 21, and 2ly be two designators of the same type in S,.

a. 21, is equivalent to 21, in (S,) =Df the sentence 21, = 21,- is true (in S,).

b. 21. is L-equivalent to 21, (in S.) =Df 2l< = 21,- is L-true (in S,).

c. 21, is F-equivalent to 21,- (in S,) =Df §li = 2[, is F-true (in S,).

No w let us see what the concepts just defined mean fm the various
kinds of d esig nators. We begin w ith predicators . Let ‘ j*(. and ‘ Q’ b e two
p redicators of deg ree one in Si._According to 3-5a, t hey are equivale nt if
and only if ‘P = Q’ is true, hence, accordina To ' 3-ib, if and only if
‘(x)Wx ^ Qy]’ is true , hence if ‘P’ ho 1H<^ for the samp i ndividuals as ‘O’.

' The result is analogous for two predicators of any degree n, say ‘ R’ and
‘R”. They are equivalent, according to 3-sa and 3-ia, if and only if

s 3. EQUIVALENCE AND L-EQUIVALENCE 15

‘ (xi) . . (**)[Ra:i . . *« = . . *»]’ is true, hence if the two predicators

hold for the same sequences (of length n) of individuals.

To give an example, let us assume the following as a biological fact :

3-6. Assumption. All human beings are featherless bipeds and vice versa.
Then the following holds:

3-7. The sentence ‘(a:) [Ha: s (F«B)a:]’ is true (in S,), but not L-true,
hence F-true.

According to 3-ib, the sentence just mentioned can be abbreviated by
‘H s F«B’. Hence, 3-5 yields:

3-8. The predicators ‘H’ and ‘F«B’ are equivalent (in S,), but not L-equiv-
alent, hence F-equivalent.

On the other hand, the truth of the sentence ‘ (a;)[Ha: = RAa;]’ can be
established without referring to facts by merely using the semantical rules
of S„ especially 1-2 (see the remark following this rule) and the truth
rules for the universal quantifier and for ‘ =’. Therefore:

3-9. ‘ (a;)[Ha: = RAa;]’ is L-true.

According to 3-ib, the sentence just mentioned can be abbreviated by
‘H = RA’:

3-10. ‘H = RA’ is L-true.

Hence, 3-5b yields;

3-11. The predicators ‘ H’ and ‘RA’ are L-equivalent (in S,).

Now let us apply our definitions to individual expressions. The follow-
ing result is obtained from 3-3 and 3- 5a;

3-12. Individual expressions are equivalent if and only if they are expres-
sions for the same individual.

Examples for L-equivalence and h'-equivalence of individual expres-
sions will be given later (§9).

A consideration of these results for predicators and individual expres-
sions shows the following : If ‘ P’ and ‘ Q’ are equivalent predicators, then
‘Pa’ and ‘Qa’ are either both true or both false and hence, in any case,
equivalent; the same holds for ‘Pb’ and ‘Qb’, etc. Furthermore, if ‘a’ and
‘b’ are equivalent, then ‘Pa’ and ‘Pb’ are either both true or both false
and hence, in any case, equivalent; the same holds for ‘Qa’ and ‘Qb’, etc.
An analogous result for functors follows from rules like 3-4. It can be
shown that the following two theorems hold generally for our systems S„
Sj, and Sj, and likewise for any similar systems, including those contain-

i6

I. THE METHOD OF EXTENSION AND INTENSION

ing functors, provided that definitions analogous to those given above are
laid down.

3-13. If two designator signs are equivalent, then any two sentences of
simplest form (in S,: atomic form) which are alike except for the occur-
rence of the two designator signs are likewise equivalent.

3-14. If two designators (which may be compound expressions) are L-
equivalent, then any two sentences (of any form whatever) which are
alike except for the occurrence of the two designators are likewise L-
equivalent.

These two results show that our choice of the interpretation for the ex-
tended uses of ‘ = ’ and of the definition for the extended use of the terms
‘equivalent’ and ‘ L-equivalent’ was not arbitrary. In fact, the choice was
made with the intention of reaching these results. In particular, the first
result 3 - 13 , in its application to individual expressions, may be regarded
as supplying a justification for the use of ‘ ’ as a sign of identity, which

might at first perhaps appear strange.

On the basis of equivalence and L-equivalence for designators we define
the following two concepts:

3-16. Definitions. Let 3Ii be a designator (in S,).

a. The equivalence class of 21, =Df the class of those expressions (in
S,) which are equivalent to 21,.

b. The L-equivalence class of 21. =d( the class of those expressions
(in Si) which are L-equivalent to 21 i.

It is easily seen that 2li itself belongs to both classes, that the L-equiva-
lence class is a subclass of the equivalence class, and that both classes con-
tain only designators of the same type as 21 ,.

§ 4. Classes and Properties

It is customary to regard two classes, say those corresponding to the predica-
tors ‘P’ and ‘Q’, as identical if they have the same elements, in other words,
if ‘P’ and ‘Q’ are equivalent. We regard the two properties P and Q as identical
if ‘P’ and 'Q’ are, moreover, L-equivalent. By the intension of the predicator
‘P’ we mean the property P; by its extension we mean the corresponding class.
It follows that two predicators have the same extension if they are equivalent,
and the same intension if they are L-equivalent. The term ‘property’ is to be
understood in an objective, physical sense, not in a subjective, mental sense;
the same holds for terms like ‘concept’, ‘intension’, etc. The use of these and
related terms does not involve a hypiostatization.

In analyzing the meaning of an adjective, e.g., ‘human’, or a cor-
responding predicator in a symbolic language, e.g., ‘H’, it is customary

i8 1. THE METHOD OF EXTENSION AND INTENSION

The above examples seem to show a certain parallelism between the two
modes of speech, the one in terms of ‘property’ and the other in terms of
‘class’. However, there is one fimdamental difference, leaving aside the
inessential, merely idiomatic difference that in the one case the connecting
phrase is ‘has’ or ‘possesses’, while in the other it is ‘belongs to’ or ‘is an
element of’. The fu ndamenta l^ difference is in the condition of identity.
Classes are usually taken as identica Tit they have the same element s.'
- TlTus, lor example, on the ba^ of our earlier assumption (3-6), the class
~^mM n ha s the same ele nTCiils as the class - Fcathcric ss Bi p ed : TheteTore :

4 - 7 . The class Human is the same as the class Featherless Biped.

roperties are usually regarded as i dentical is less
clear. It seems natural, and sufficiently in agreement with the vague”
-t^Istornary usag^ to ’regard" properties as identical iT'il (laii be Sliuwn by
logical means alone, without reference to facts, that whateveFffas^e one
property has the other and vice versa ;-m other words^ iLtbe, equivalence
seiitence is not only true but L-true. Thus with respect to our earlier ex-
amples (3-7 and 3-9) the following holds:

4 - 8 . The property Human is not the same as the property Featherless
Biped.

4 - 9 . The property Human is the same as the property Rational Animal.

It is easily seen, on the basis of our definitions in the preceding section
(3-ib and 3-5a, b) that the identity conditions stated above can be formu-
lated in the following way with respect to predicators (of degree one) :

4 - 10 . Classes are identical if and only if predicators for them are equiva-
lent.

4 - 11 . Properties are identical if and only if predicators for them are
L-equivalent.

Now we shall introduce the terms ^extension' and 'intension* with
respect to predicators. If two predicators apply to the same individuals —
in other words, if they are equivalent — it is sometimes said that they are
coextensive or that they have the same extension (in one of the various
customary uses of this term) . The use of ‘ intension’ varies still more than
that of ‘extension’. It seems in agreement with at least one of the custom-
ary usages to speak of the same intension in the case of L-equivalence.
Thus we lay down the following two conventions:

4 - 12 . Two predicators have the same extension if and only if they
are equivalent.

§ 4. CLASSES AND PROPERTIES 19

4 - 13 . Two predicators have the same intension if and only if they
are L-equivalent.

These conventions determine only the use of the phrases ‘have the same
extension’ and ‘have the same intension’. For many purposes this is suf-
ficient. If, however, we wish to go further and to speak of something as the
extension of a given predicator, and of something else as its intension,
then these conventions do not suffice; but they help us by narrowing the
choice of suitable entities. The first convention means that we may take
as extensions of predicators only something which equivalent predicators
have in common. According to 4-10, this condition is fulfilled by the cor-
responding classes. The second convention means that we may take as in-
tensions of predicators only something which L-equivalent predicators
have in common. According to 4-11, this condition is fulfilled by the cor-
responding properties. This suggests the following conception of the ex-
tension and the intension of predicators:

4 - 14 . The extension of a predi cator (of degree o ne) is the corresponding
class.

4 - 16 . The intmsion^of a predicator (of degree one) is the corresponding
prop^ty.

This seems sufficiently in agreement with customary usage. If this is ap-
plied to the predicator ‘H’ in S„ we obtain:

4 - 16 . The extension of ‘H’ is the class Human.

4 - 17 . The intension of ‘H’ is the property Human.

Both results hold also for the predicator ‘(Xx)(Hx)’, which is L-equiva-
lent to ‘H’ in S,.

It is obvious that there are many other ways for choosing entities as ex-
tensibns andlhtensions of predicators (of degree one) so as to satisfy our
c onventions t4-r2 and 4-13 ). One alternative is as follows: It is possible
to take as the extension of a predicator Jts ^uivalence class .(^-15^) and
as i ffi intension its L-equivalenceclass (3-1 sb). This conception seems less
natural than the one we have chosen (4-14, 4-15), because it leads to
linguistic instead of to extra-linguistic entities. On the, other hand, this
conception~ 6 rm t 5 Ti 5 ions ha^the advantage that it is possible in an exten-
sional metalanguage; this will be explained later. (Compare definitions by
Russell and Quine^nehtioned below, at the end of § 33.)

It may perhaps be useful, in order to avoid misunderstandings, to add
some informal remarks concerning the use in this book of the term
^property'. This term will be used as synonymous \dth words like Iqual^
it y’, ‘charact^ ‘character istic’, and the like in their ordinary use., It

20

I. THE METHOD OF EXTENSION AND INTENSION

is to be understood in a very wide ^nse, including whatever can be said
ISeahingfully, no matter whether truly or falsely, about any individual.
"The tenhls usecf here not only for qualitative properties in the narrower
sense (for example, the properties Blue, Hot, Hard, and the like) but also
for quantitative properties (for example, the property Weighing Five
Pounds), for relational properties (e.g., the property Uncle Of Some-
body), for spatiotemporal properties (e.g., the property North Of
Chicago), and others. It is important to note what is noi meant here by the
term ‘property’. First, it does not jefer lo_linguistic-expigssions; to the
symbol ‘H’ and the corresponding word ‘human’ we apply the term
‘ predicator’, not ‘property’; by a pro perty we mean rather what is ex-
pressed by a predicator (of degree one). Secqndj_ the properties of things
are not meant as something mental, s^ images or sense-data, but as
something physical that the things have, a side or aspect or £omponent
onshaiacter of the things. If an observer sees that this table is red, then
the table has the character Red and the observer has the corresponding
character Red-Seeing. By the property Red we mean the first, not the
second; we mean that physical character of the thing which the physicist
explains as a certain disposition to selective reflection, not that psycho-
logical character of the observer which the physiologist explains as a
certain disposition to a specific reaction by the sensory part of the nervous
system.

Suppose we understand some predicators in a given language; that is to
say, we know which properties they express. Suppose, further, that we
have experienced each of these properties; that is to say, we have, for each
of them, found some things which, according to our observations, have
that property. We can form compound predicators out of the given predi-
cators with the help of logical particles. Then we understand a compound
predicator because its meaning is det ermined by tne mea nings ol the com-
ponent p redicators an d the logical structure of the compound expression,
i t is Important to ^w tire thsFnhf nnTIpi^tarifhn g pf a compound predic a-
tor IS no longer dependent upon o bserva tions of any things to .which it ,
applies, that is, a ny th ings which have ^e complex jjroperty expressed
by if. ~

TiTdrder to construct examples, suppose that the system S, contains
not only the predicator ‘H’ for the property Human, but also the predica-
tor ‘T’ for the property Twenty Feet High. Then we can, for example,
form the following compound predicators (provided we permit the use of

’ and ‘ V’ in predicators in analogy to the use of ‘ • ’ introduced by 3-2) :
‘~H’ expresses the property Non-Human, ‘H V T’ the property Human

§ 4. CLASSES AND PROPERTIES

21

Or Twenty Feet High, and ‘H*T’ the property Human And Twenty Feet
High. We know things which exemplify the first of these three properties,
and likewise some for the second. But we have never seen any things that
exemplify the predicator ‘ H*T’, and there are presumably no things of this
kind in the world. Nevertheless, ‘H*T’ is not meaningless. Since it is a
well-formed predicator {of degree one), it expresses a property, although
this property does not apply anywhere. We shall say of both the predica-
tor and the property that they are empty. One can understand ‘H«T’ just
as clearly as the other compound predicators; and one may indeed under-
stand this or any other compound predicator before he knows whether
and, if so, where it is exemplified. The understanding of a compound
predicator is based upon the understanding of the component predica-
tors. Exemplification in experience is required only for primary predica-
tors, with the help of which the others are interpreted.

Now consider the predicator ‘ H* '^H’. No factual knowledge is needed
for recognizing that this predicator cannot possibly be exemplified. Never-
theless, this expression is not meaningless. It is a well-formed predicator;
it expresses the property Human And Non-Human." We shall say of
both the predicator and the property that they are L-empty (logically
empty). [There is only one L-empty property, although there are many
empty properties. If ‘P’ and‘Q’ are any two L-empty predicators, then
‘P = Q’, that is, ‘{x)(Px = Qx)’ (3-ib), is L-true; therefore, ‘P’ and ‘Q’
are L-equivalent (,3-5b); hence they express the same property (4-1 1).]

The use of the term 'relation' in this book is analogous to that of the
terrir^pr oper ty ^qust explained. A relation is meant neither as a mental
efiliiyTTOias air expression but rather as something that is expressed by
certain designators, namely, predicators of degree two or more, and that
may hold objectively for two or more things.

The term 'concept' will be used here as a common designation for
properties, relatTohs, and similar entities (including individual concepts, to
be explained in 1 9, and f uncti ons, but not propositions). For this term it
is especially important to stress the fact that it is not to be understood in
a mental sense, that is, as referring to a process of imagining, thinking,
conceiving, or the like, but rather to something objective that is found in
nature and that is expressed in language by a designator of nonsentential
form. (This does not, of course, preclude the possibility that a concept —
for example, a property objectively possessed by a given thing — may be
subjectively perceived, compared, thought about, etc.)

" Compare Bennett and Bayiis, [Logic], sec. 3.4: “The existence of self-inconsistent
concepts."

22

I. THE METHOD OF EXTENSION AND INTENSION

The preceding remarks are meant merely as an informal terminological
clarification. They should by no means be regarded as an attempt toward
a solution of the old controversial problem of the universals. The tradi-
tional discussions concerning this problem are, in my view, a rather
heterogeneous mixture of different components, among them logical state-
ments, psychological statements, and pseudo-statements, that is, expres-
sions which are erroneously regarded as statements but do not have cogni-
tive content, although they may have noncognitive — for instance, emo-
tive — meaning components. My remarks on the interpretation intended for
the term ‘property’ are admittedly rather vague, chiefly because of a lack
of a clear and generally accepted terminology about matters of this kind.
Nevertheless, I hope they will give sufficiently clear indications for all
practical purposes and, above all, may help to avoid certain typical mis-
understandings.

I wish to emphasize the fact that the discussions in this book about
properties, and similarly about relations, concepts in general, proposi-
tions, etc., do not involve a hypostatization . As I understand it, a hypos-
tatizatio n or substantializati on or reific atipiL tonsists in^ mistaking as
things entities~wKicK~are not things. Examples of hypostatizations of
properties (or ideas, universals, or the Tike) in this sense are such formula-
tions as ‘the ideas have an independent subsistence’, ‘they reside in a
super-heavenly place’, ‘they were in the mind of God before they became
manifested in things’, and the like, provided that these formulations are
meant literally and not merely as poetical metaphors. (We leave aside
here the historical question of whether these hypostatizations are to be
attributed to Plato himself or rather to his interpreters.) These formula-
tions, if taken literally, are pseudo-statements, devoid of cognitive con-
tent, and therefore neither true nor false. Whatever is said in this book
about properties may be wrong, but it has at least cognitive content. This
follows from the fact that our statements belong to, or can be translated
into, the general language of science. We use the term ‘property’ in that
sense in which it is used by scientists in statements of the following form:

‘ These two bodies have the same chemical properties, but there are certain
physical properties in which they differ’; ‘Let us express the property . . . ,
which is exemplified by the one but not by the other of these two bodies,
by‘P”.

The terrn ‘entity’ is frequently used in this book. I am aware of the
metaphysical connotations associated with it, but I hope that the reader
will be able to leave them aside and to take the word in the simple sense in
which it is meant here, as a common designation for properties, proposi-

§ 5. EXTENSIONS AND INTENSIONS

*3

tions, and other intensions, on the one hand, and for classes, individuals,
and other extensions, on the other. It seems to me that there is no other
suitable term in English with this very wide range.

§ 6. Extensions and Intensions

In analogy to the case of predicators, we shall say of two designators of any
kind that they have the same extension if they are equivalent, and that they
have the same intension if they are L-equivalent. In later sections we shall dis-
cuss the problem of finding suitable entities which might be taken as extensions
and intensions in accordance with these identity conditions. If two predicators,
say ‘P’ and ‘Q’, are equivalent or L-equivalent in a system S, we say also that
the properties P and Q are equivalent or L-equivalent, respectively; and anal-
ogously with designators of other kinds and their intensions.

In the preceding section we introduced the terms ‘extension’ and ‘inten-
sion’ with respect to predicators only, in agreement with traditional us-
age. Now we shall extend the use of these terms, applying them to other
types of designators in an analogous way.

In the case of predicators, we have taken equivalence as the condition
for identity of extension, and L-equivalence for identity of intension (4-12
and 4-13). Earlier (§ 3), we saw how the semantical concepts of equiva-
lence and L-equivalence can be applied to the various types of designa-
tors. Thus it seems natural to take the same conditions as defining identity
of extension or intension with respect to designators in general. This leads
to the following definitions; 4-12 and 4-13 are now regarded simply as
special cases hereof.

6 - 1 . Definition. Two designators have the same extension (in S,) =Df
they are equivalent (in S,).

6 - 2 . Definition. Two designators have the same intension (in S,) =d(
they are L-equivalent (in S,).

Note that the terms ‘extension’ and ‘intension’ have not been defined
hereby, but only the phrases ‘ have the same extension’ and ‘ have the same
intension’. In order to speak about extensions and intensions themselves,
we have to look for entities, or at least for phrases apparently referring
to entities, which can be assigned to designators in accordance with these
definitions. In the case of predicators, we found classes and properties as
such entities. We shall see later how suitable entities can be chosen for
sentences and individual expressions.

The introduction into the metalanguage M of expressions for additional
kinds of entities is always a precarious step that must be taken with cau-
tion and with careful consideration of the consequences. We shall discuss
the problem involved in the introduction of extensions and intensions for

24

I. THE METHOD OF EXTENSION AND INTENSION

designators later (§§ 33 ff.). Here it may be noted that the phrases ‘have
the same extension’ and ‘have the same intension’, although apparently
referring to certain entities as extensions and intensions, are, in fact, en-
tirely free of the problematic nature of the terms ‘extension’ and ‘inten-
sion’ ; for those phrases are based by the above definitions on the terms
‘equivalent’ and ‘L-equivalent’, and these go back (by 3-5) to the terms
‘true’ and ‘L-true’, which can be defined for the system Si in an exact
way, as explained earlier.

It is often convenient to apply the term ‘equivalent’, and perhaps also
the term ‘ L-equivalent’, not only to designators but likewise to the inten-
sions of these designators; thus not only to predicators (e.g., ‘the predica-
tors ‘H’ and ‘F«B’ are equivalent in S/) but also to properties and rela-
tions (e.g., ‘the property Human and the property Featherless Biped are
equivalent’) ; and analogously not only to sentences but also to proposi-
tions. This transferred use cannot lead to any actual ambiguity or con-
fusion, for two reasons: (i) The context always makes clear whether the
term ‘equivalent’ is meant in the original or in the transferred sense; the
former is the case whenever the term is applied to expressions in a lan-
guage system, the latter whenever it is applied to in tensions, hence to extra-
linguistic entities. (2) In the original use the term is accompanied by a
reference to a language system (e.g., ‘ equivalent in S/ ; however, this holds
only for the complete formulation; in practice we often omit the reference
if the context makes clear which language system is meant) ; the trans-
ferred use is not so accompanied (e.g., it makes no sense to say ‘these two
properties are equivalent in S,’). Two designators may be equiva lent in
one language and not in another,Tecause they may have other rneanings
in the second language ; thus the equivalence of designators is dq^ndenj; .
upon the l anguage, as all sema ntical concepts are. On the other hand_^ Ae
equivalent of two pro pertie s is not dependent upo n langu age; it is a non-
semantical and, moreover, a nonlinguistic conce pt ( e.g., i t is a biological,
not a linguistic, fact that the property Human an d the property Feather-
less Biped are e quival ent). The term ‘equivalent’ in the transferred use
stiff belongs to the metalanguage M; not, however, to the semantical
part of M but to what we might call the object part, that is, that part of
M which contains the translations of the sentences and other expressions
of the object languages." The application of the terms ‘equivalent’ and

" Previously, I called terms of this kind, which are transferred from semantics to extra-
linguistic entities, absolute terms ([I], § 17), in order to indicate that in their new use the
terms are no longer relative to a language. However, I now prefer to avoid the word ‘absolute’
because some readers were puzzled by it and suspected behind it some sort of metaphysical
absolutism.

§ 6. EXTENSIONS AND INTENSIONS OF SENTENCES 25

‘ L-equivalent’ to intensions of designators, if these designators are equiva-
lent or L-equivalent in the original semantical sense, leads, in combination
with the identity conditions expressed in 5-1 and 5-2, to the following
results:

6 - 3 . If two designators are equivalent (in S,), then we say that their ex-
tensions are identical and that their intensions are equivalent.

6 - 4 . If two designators are L-equivalent (in S,), then we say that their in-
tensions are L-equivalent (or identical).

“because of 5-3, ‘ s’ may be regarded as a sign both for the identity of
extensions and for the equivalence of intensions; in particular, if it stands
between predicators of degree one (as in ‘H s F»B’), it is a sign of identity
of classes and a sign of equivalence of properties.

Examples. We found earlier that the predicators ‘H’ and ‘F#B’ are
equivalent but not L-equivalent (3-8), and that ‘H’ and ‘RA’ are L-
equivalent (3-1 1). If we apply here the above two definitions, we obtain
the following formulations with transferred terms, in addition to the
earlier formulations in terms of identity (4-7, 4-8, and 4-9) ;

6-6. The property Human is equivalent to the property Featherless
Biped.

6-6. The property Human is not L-equivalent to the property Featherless
Biped.

6 - 7 . The property Human is L-equivalent to the property Rational
Animal.

Of these three formulations, only the first is actually useful in M ; the
other two serve only as preparation for analogous formulations in another
metalanguage M' (§ 34). [It may be remarked incidentally that the terms
‘equivalent’ and ‘L-equivalent’ in their transferred, nonsemantical use,
which are here applied to intensions, could also Be applied to extensions.
However, equivalence of extensions would be the same as identity of
extensions and hence would not be useful. And to speak of L-equivalence
of extensions would even be dangerous because it would lead to the same
consequences that we shall later find for sentences like 42-6A.]

§ 6. Extensions and Intensions of Sentences

We take as the extension of a sentence its truth-value, and as its intension
the proposition expressed by it. This is in accord with the identity conditions
for extensions and for intensions stated in the preceding section. Propositions
are here regarded as objective, nonmental, extra-linguistic entities. It is shown
that this conception is applicable also in the case of false sentences.

26 I. THE METHOD OF EXTENSION AND INTENSION

Now let us see whether we can find entities which may be taken as
extensions and intensions of sentences in accordance with our definitions
for the identity of extensions (5-1) and of intensions (5-2).

According to 5-1, we must take as extensions of sentences something
that equivalent sentences have in common. The most natural choice seems
the truth- values:

6 - 1 . The extension of a sentence is its truth-value.

At first glance, it may perhaps seem strange to call a truth-value an
extension, and perhaps there may be a feeling even against saying that
equivalent sentences have the same extension. The term ‘extension’ seems
natural enough in the case of predicators; we easily visualize the domain of
individuals as an area and the class of individuals to which a certain predi-
cator applies (e.g., the class Biped for the predicator ‘B’) as a subarea
which extends over a smaller or larger part of the whole. But one might
say that in the case of a truth-value there is nothing extended. However,
a closer inspection may remove the impression of strangeness. It has be-
come customary to use the term ‘ cxtensional’ for truth-functional connec-
tions, i.e., for connections such that the truth-value of the full sentence is
a function of the truth-values of the components. And there is, indeed,
a strong analogy between truth-values of sentences and extensions of
predicators. This can be seen as follows: A predicator of degree n is char-
acterized by the fact that we must attach to it n argument expressions in
order to,_form a sentence. Therefore, a sentence might be regarded as a
predicator of degree zero. Let ?(. and 31 , be any predicators of degree
« ^ i); then (according to 4-12, 3-5, and 3-1 a) 31 . and 31 , have the

same extension if and only if (x,)(x,)- . . (x„)[3L2:,x, . , = 31 , x,*:, . . xf[

is true. If we stipulate that this, which applies originally only to w ^ i,
is to be applied analogously to sentences as predicators of degree zero,
we find that two sentences, ©, and ®„ have the same extension if and
only if s ©, is true, hence if and only if S. and @, are equivalent.
Thus we are led back to 5-1 as applied to sentences; and then it seems
natural to take the truth-values as extensions. [For the time being we may
leave aside the question of what kind of entities these truth-values are,
which are here proposed as extensions. This problem will be discussed
later (in § 23).]

Now we have to decide what entities to take as intensions of sentences.
It is often said that a (declarative) sentence expresses a proposition. We
accept this use of the word ‘ proposition’ ; that is to say, we do not use this
word for sentences or for sentences together with their meaning b ut for

§ 6. EXTENSIONS AND INTENSIONS OF SENTENCES

27

those entities which themsel ves are extra-linguistic but which, if they find
expressio n in a language, are express ed by (det^faratryeri^tences.*^ Those
'authors who use the term ‘proposition’ in this'‘senselLre"oTfen not quite
clear as to the condition under which two sentences express the same
proposition. We decide to take L-equivalence as this condition.'^ Thus,
for example, we say that the sentences ‘~(Pa»QbV and ‘~PaV~Qb’
express the same proposition. This seems sufficiently in agreement with
the usage of many logicians. Since we took L-equivalence as the condition
of identity for intensions ( 5 - 2 ), we may regard propositions as intensions:

6-2. The intension of a sentence is the proposition expressed by it.

Examples:

6-3. The extension of the sentence ‘Hs’ (in S,) is the truth-value that
Scott is human,'® which happens to be the truth.

6-4. The intension of the sentence ‘Hs’ is the proposition that Scott is
human.'®

Some remarks may help to clarify the sense in which we intend to use
the term ^proposition' . Like the term ‘property’ (§ 4 ) , it is used neither
f or a li aflui&tk axpi fission nor for a subjective, mental occurrence, but
rather for so mething objective tha t may or may n ot be exem piffled in'
nature. [ We might say that proposi tions are, like j^operties, o£a co ncep-
tual nature . But it may be better to avoid this formulation, because it
might lead to a subjectivistic misinterpretation, if the fact is overlooked
that we use the term ‘concept’ in an objective sense (see § 4 ).] We apply
the term ‘ proposition’ to any entities of a certain logical type, namely,
those that may be expressed by (declarative) sentences in a language.
By the property Black we mean something that a thing may or may not
have and that this table actually has. Analogously, by the proposition that
this table is black we mean something that actually is the case with this
table, something that is exemplified by the fact of the table’s being as it is.

‘3 On the necessity of distinguishing clearly between the two meanings of the term ‘propo-
sition’, compare [I], pp. 235 f.

Compare [I], p. 92.

‘s In analogy to ‘the property Human’ and ‘the class Human’ we might write here ‘the
proposition Scott-Is-Human’ and ‘the truth-value Scott-Is-Human’. However, this would
become rather awkward for longer sentences. Therefore, we shall instead insert ‘that’ after
‘proposition’, thus coming back to ordinary usage. For the sake of analogy, we shall likewise
write ‘the truth-value that . . .’, although it deviates from ordinary usage; we cannot use the
more idiomatic form ‘the truth-value of the proposition that . . .’ because in 6-3 we wish to
speak only about the sentence and its extension, the truth-value, not about its intension, the
proposition.

38

I. THE METHOD OF EXTENSION AND INTENSION

(This simple explanation is possible only in the case of a true proposition ;
the problem of false propositions will soon be discussed.)

'^The question of whether facts are propositions of a certain kind or
entities of a different nature is controversial. Ducasse'* identifies facts with
true propositions. Bennett and Baylis*’ say that propositions are true or
false; on the other hand, “facts themselves are neither true nor false,
but just are”. The question is, to a certain extent, a terminological one and
hence to be settled by convention. Since the term ‘fact’ in its ordinary
use is rather vague and ambiguous, there is some freedom of choice left as
to how to turn it into an exact technical term, in other words, how to
explicate ifyTam inclined to think, like Ducasse, that it would not deviate
too much from customary usage if we were to explicate the term ‘fact’
as referring to a certain kind of proposition (in our objective sense of
this ^^fd). What properties must a proposition have to be a fact in this
sense? First, it must, of course, be true; second, it must be contingent (or
factual) ;^us it must be F-true. I think that still another requirement
should be added: The proposition must be specific or complete in a certain
sense; but I am not sure what degree of completeness should be required.
An example may illustrate the problem. The proposition that this thing
(a piece of paper I have before me) is blue is a true proposition; in other
words, this thing has the property Blue. But the property Blue has a wide
range; it is not specific but includes many different shades of blue, say
Blue,, Blue,, etc. This thing, on the other hand, or, more exactly speaking,
a specified position c on its surface at the present moment has only one
of these shades, say Blucj . Let p be the proposition that c is blue, and q
the more specific proposition that c is bluej. It is the truth of q that makes
p true. Therefore, the nonspecific proposition p should perhaps not be
regarded as a fact. Whether q should be so regarded remains doubtful ; q is
completely specific in one respect, concerning the color, but it does not
specify the other properties of the given thing^’^Should we require com-
plete specificity with respect to all properties of the thing or things in-
volved, and also with respect to all relations among the given things, or-
perhaps even with respect to all relations between the given things and all
other things? It seems somewhat arbitrary to draw a line at any of these
points. If we do not stop at some point but go the whole way, then we
arrive at the strongest F-true proposition pi, which is the conjunction of

C. J. Ducasse, “Propositions, Opinions, Sentences, and Facts”, Journal of Philosophy,
XXXVII (1940), 701-1 1 ; see also his reply to some objections {ibid., XXXIX [1942], 132-36).

[Logic], p. 49.

g 6. EXTENSIONS AND INTENSIONS OF SENTENCES

29

all true propositions'* and hence L-implies every true proposition. If we
require of a fact this maximum degree of completeness (short of L-falsity),
then there is only one fact, the totality of the actual world, past, present,
and future. We indicate here these various possibilities for choosing an
explicatum for the concept of fact without making a decision. We shall
not take the term ‘ fact’ as a technical term but shall use it only in informal
explanations; thus, for example, we have said (§ 2) that the truth-value
of a sentence which is not L-determinate is dependent upon the facts.

The greatest difficulty in the task of explicating the concept of proposi-
tion is involved in the case of a false sentence. Since this piece of paper c
is, in fact, blue, sentences like ‘c is not blue’ or ‘c is red’ are false. They
cannot be regarded as meaningless, because we understand their meaning
before we know whether they are true or false. Therefore, these sentences,
too, express propositions. On the other hand, these propositions cannot
have the same relation to facts as the proposition expressed by the true
sentence ‘c is blue’. While the latter proposition is exemplified by a fact,
the former ones are not. What, then, are these false propositions? Are
there any entities of which we can say that they are expressed by those
false sentences, but for which we cannot point out any exemplifying facts?

Russell has given a thorough discussion of the problems here involved.
He likewise decides to use the term ‘ proposition’ for what is expressed by a
sentence, in other words, for the signification of a sentence, provided that
an entity of this kind can be found. But he despairs of finding an entity
of this kind in the objective, factual realm. He argues as follows: “Since
a significant sentence may be false, it is clear that the signification of a
sentence cannot be the fact that makes it true (or false). It must, there-
fore, be something in the person who believes the sentence, not in the ob-
ject to which the sentence refers.”'” “Propositions . . . are to be defined
as psychological and physiological occurrences of certain sorts — complex
images, expectations, etc. . . . Sentences signify something other than
themselves, which can be the same when the sentences differ. That this
something must be psychological (or physiological) is made evident by
the fact that propositions can be false.”'® Thu s i t see ms that Russell
chooses a subjective, mental PYpHri^tji m fpr thp nl-prnppgitiop

only or maiHIy Tor" the reason that, in his opinion, there is no other way nf ..

TOnnech^jKithLfalse pioposiJtions.

'* For the concepts of disjunctions or conjunctions of infinitely many propositions see [I],
pp. 93 f.

” Russell, [Inquiry], p. 229 (chap, xiii, in sec. A). (Page numbers refer to the American
edition; it seems that the British edition has, unfortunately, a different pagination.)

Ibid., pp. 237 f. (chap, xiii, end of sec. A).

30

I. THE METHOD OF EXTENSION AND INTENSION

I believe that it is possible to give an objective interpretation to the
term ‘proposition’, which is still applicable in the case of false sentences.
Any proposition must be regarded as a complex entity, consisting of com-
ponent entities, which, in their turn, may be simple or again complex.
Even if we assume that the ultimate components of a proposition must be
exemplified, the whole complex, the proposition itself, need not be. The
situation can perhaps best be made clear by its analogy with the situation
concerning properties. As we have seen earlier (§ 4), a compound predica-
tor, for example, ‘H»T’, may express an empty property, that is, one not
exemplified by any individual. The components ‘H’ and ‘T’ express
properties which are exemplified. The property expressed by the com-
pound predicator is constituted out of the component properties in a
logical structure indicated by the logical particles connecting the com-
ponent predicators. Thus we see that the fact that some predicators are
empty cannot prevent the explication of properties as objective entities.
Analogously, the fact that some sentences are false does not exclude the
explication of propositions as objective entities. Propositions, like com-
plex properties, are complex entities; even if their ultimate components
are exemplified, they themselves need not be. The difference between
propositions and complex properties or other complex concepts is merely a
difference in the logical type. Therefore, the kind of connection is different.
In the case of our example ‘H«T’, the connection was that of conjunction.
There are other logical connections which, applied to nonpropositional
components of suitable types, result in propositions. Consider as an ex-
ample the sentence ‘ Hs’ of the system S,; it consists of the predicator ‘ H’
and the individual constant ‘s’ combined by juxtaposition. Therefore, it
expresses a complex intension of propositional type. Its two components
are the intension of ‘H’, which is the property Human, and the intension
of ‘s’, which is, as we shall see later (§ 9), the individual concept Walter
Scott. The logical connection of these two intensions is that of attribution
or predication (expressed in S, simply by juxtaposition; its converse is
expressed in certain other symbolic languages by ‘ e’ and in English by the
copula ‘is’). Thus the resultant intension of the sentence is the proposition
that Scott is human. As an example of a different structure' take
‘(a:)(Bx D Fx)’. The intension of ‘B’ is the property Biped, that of ‘F’
is the property Featherless. These two properties are the components of
the complex intension of the whole sentence. They are connected by the
universal conditional connection, expressed, according to the rules of the
system, by the way in which ‘ B’ and ‘ F’ are combined in the sentence with
the help of three occurrences of a variable, two pairs of parentheses, and

§ 6. EXTENSIONS AND INTENSIONS OF SENTENCES

31

the conditional connective ‘ D This kind of connection yields, if applied
to two properties, a proposition. Thus the complex intension expressed by
the sentence is the proposition that whatever is a biped is featherless.
Each of the two component properties is exemplified by some individuals.
Some of the sentences of the form ‘ Bx D Fa:’ are exemplified by facts and
hence true, but some of them are not. The whole intension is not exempli-
fied; but it is, nevertheless, a proposition because it consists of exemplified
components in a propositional structure; just as the intension of ‘H«T’,
though empty, is a property because it consists of two exemplified com-
ponents in the structure of a property. Thus F-false sentences, too, express
propositions. Now we may go one step further. Consider the L-false
sentence ‘(H*~H)s’. It consists of the predicator and the

individual constant ‘s’, in the same combination as in the previous ex-
ample ‘Hs’. We have seen earlier (§4), that the predicator
although L-empty, expresses a property, namely, the L-empty property
Human And Non-Human. Therefore, the sentence mentioned expresses a
complex intension resulting from combining this property with the indi-
vidual concept Walter Scott by attribution. Thus this intension is the
proposition that Scott is human and not human. Although this intension,
like that of ‘H*~ H’, cannot possibly be exemplified, it still is a proposi-
tion. By going one step further in the analysis of this proposition we find
as its components the property Human and the individual concept Walter
Scott; these components are both exemplified, and they are combined in a
structure of propositional typ<

Generally speaking, it mu st
primarily express an intension only if it is exemplified. However, once we'
have some designators which have a primary intension, we can build com-
pound designators out of them which express derivative, complex inten-
sions, no matter whether these compound designators are exemplified or
noty^e do not need exemplifications in order to grasp their intensions,
because the intension of a compound designator is determined, in virtue
^f the semantical rules of the system, by the intensions of the component
designators and by the way in which these designators are combined.

It has been the purpose of the preceding remarks to facilitate the under-
standing of our conception of propositions. If, however, a reader should
find these explanations more puzzling than clarifying, or even unac-
ceptable, he may simply disregard them. They are not a necessary basis
for the further discussions in this book; they will hardly be referred to
again. It will be sufficient for nearly all our discussions involving proposi-
tions to assume that they are entities of any kind fulfilling the following

32

I. THE METHOD OF EXTENSION AND INTENSION

two conditions: (i) to every 'Sentence in a semantical system S, exactly
one entity of this kind is assigned by the rules of S; (2) the same entity
is assigned to two sentences in S if and only if these sentences are L-
equivalent. If someone is in doubt as to whether there are any nonmental
and extra-linguistic entities which fulfil these conditions, he may take as
propositions certain linguistic entities which do so. We shall later see that,
for example, certain classes of sentences in 5 may be taken (the L-equiva-
lence classes, see remark at the end of § 33) or certain classes of classes of
sentences in 5 (the ranges, see remark near the end of § 40).

§ 7. Individual Descriptions

An (individual) description is an expression of the form ‘(ix)(. . x . it
means ‘the one individual such that If there is one and only one

individual such that . . x . . , we say that the description satisfies the unique-
ness condition. In this case the descriptum, i.e., the entity to which the descrip-
tion refers, is that one individual. Logicians differ in their interpretations of de-
scriptions in cases where the uniqueness condition is not satisfied. The methods
of Hilbert and Bernays and of Russell are here discussed, that of Frege will be
discussed in the next section.

We use the term Hndividua V not for one particular kind of entity but,
rather, relative to a language system S, for those entities which are taken’
as the dements of the universe of discourse in S, in other words, the ej^i-”
ties of lowest level (we call it level zero) dealt with in S, no matter w hat
these entities are. For one system the individuals may be physical things,
for another space-time points, or numbers, or anything else. Consequent-
ly, we call the variables of level zero individual variables, the constants
individual constants, and all expressions of this level, whether simple
(variables and constants) or compound, individual expressions. The
most important kinds of compound individual expressions are: (i) full
expressions of functors (e.g., ‘3 -f- 4’, where ‘ -f’ is a functor and ‘3’ and
‘ 4’ are individual constants) ; within our systems, expressions of this kind
occur only in S3, not in S, and S^; (2) individual descriptions. We shall use
here the term ‘description’ mostly in the sense of ‘ individual description’.
Descriptions of other types do not occur in our systems; a few remarks on
them will be made at the end of § 8.

A description in S, has the form ‘(ix)(. . a: . it is interpreted as
‘the one individual x such that . . x . ‘{ix)' is called an iota-operator;

the scope ‘ . x . .’ is a sentential matrix with ‘x’ as a free variable. For
example, ‘ (ix) (Px» •mean»-+he'gaHie as'*ThE“6ne"indivTduarv^ch
is P and not Q’.

The entity for winc h a description stands (if there is such an entity^ will

s 7. INDIVIDUAL DESCRIPTIONS

33

be called its detcrijtt um; here, in the case of in dividual descriptions, the
'^ ^criptum is an individual . With respect to a given description, thereHJe”
'Iwo possible cases: either (i) there is exactly one individual which fulfils
the condition expressed by the scope, or (2) this does not hold, that is,
there are none or several such individuals. In the first case we shall say of
the scope, and also of the whole description, that it satisfies the unique-
ness condition :

7-1. Definition. Let . a: . be a (sentential) matrix (in S,) with ‘a:’ as
the only free variable. . x . (and ‘(ix) (. . x . satisfies the unique-
ness condition (in S,) =Df ‘(32)(a;)[. . x . . =(x = z)]’ is true (in S,).
(‘x ^ z' means ‘x is the same individual as z’; see 3-3.)

In the case of a description satisfying the uniqueness condition, there is
general agreement among logicians with respect to its interpretation ; the
one individual satisfying the scope is taken as descriptum. In the other
case, however, there is, so far, no agreement. Various methods have been
proposed. We shall outline three of them, those proposed by Hilbert and
Bernays (I), Russell (II), and Frege (III). Then we shall adopt Frege’s
I method for our systems. It shou ld be noticed that the various conceptions
mow to be discussed are not to be understood as different opinions, so that
at least one of them must be wrong, but rather as different proposals. The
different i nterpretations of descriptions are not meant as assertions about
t^e meaning of phrases of the form ‘the sp^and-so' in English^ but as pro-
poses for an interpretation an d, consequently , ior.rieductive rules, con-
cerning descriptions in symbolic systems. Therefore, there is no theo-
retical issue of right or wrong between the various conceptions, but only
fhe practical question of the comparative convenience of different
methods.

In order to make the following discussions more concrete, let us suppose
that two (sentential)_matrices are given^each with exactly one free vari-
able; we indic ate them here with the help of dots and dashes: ‘ . x .
and‘- - -’ (e.g^, ‘Axw’ and ‘HjQ.^e construct the description with the
first jLS scope and substitute it fo r ‘ v^*in thej ^dhd : * ”

7-2. - {ix)(. . X . (^Example: ‘H( ia:) (A aw)

Method I. Hilbert and Bernays,” in a system with natural num-
bers as individuals, permit the use of a description only if it satisfies
the uniqueness condition. Since the system is constructed as a calculus,
not as a semantical system, the formula of uniqueness is required to be
C-true (provable) instead of true. It seems that this method is quite con-

” [Grundlagen I], p. 384.

34

I. THE METHOD OF EXTENSION AND INTENSION

venient for practical work with a logico-arithmetical system^ one uses
ajicscription only after he has proved the uniqueness. HoweverT^is
method has a serious disadvantage, although of a chiefly theoretical
nature: the rules of formation become inde finite, i.e., there is no general
procedure for determining whether any given expression of the form 7-2
is a sentence of the system (no matter whether true or false, provable or
not). For systems also containing factual sentences, the disadvantage
would be still greater, because here the question of whether a given ex-
pression is a sentence or not would, in general, depend upon the con-
tingency of facts.

Method n. RusseW^ takes the whole expression 7-2 in any case as a
sentence. The uniqueness condition is here taken not as a precondition for
the s ^nt^ntial character of the expression but rather as one of the condi~
tinns f o r its truth — in other words, as part of its content. Thus the transla-
tion of 7-2 into M is as follows;

7 - 3 . ‘ There is an individual y such that y is the only individual for which
. . y . . holds, and - - y - (for example, ‘ there is an individual y such that
y is the only individual which is an author of Waverley, and y is human’)-

Hence, 7-2 is here interpreted as meaning the same as the following (with
a certain restriction, see below) :

7 - 4 . ‘(3y)[(2:)(. . 2; . . = (x = y)) • - - y - (In the example,

‘ (3y) [(2:) (A2 :w = (x = y)) • Hy] ’.)

In order to incorporate this interpretation into his system, Russell lays
down a contextual definition for descriptions; 7-2 is the definiendum, 7-4
the definiens. If we prefer to take the iota-operator as primitive instead of
defining it, we can reach the same result by framing the semantical rules
in such a way that any two sentences of the forms 7-2 and 7-4 become
L-equivalent.

In comparison with Hilbert’s method, Russell’s has the advantage
that an expression of the form 7-2 is always a sentence. In comparison
with Frege’s method, which will soon be explained, it has the disad-
vantage that the rules for descriptions are not so simple as those for other
individual expressions, especially those for individual constants. In par-
ticular, the inferences of specification, leading from ‘ (y){- - y - to
‘--a--’, and of existential generalization, leading from ‘--a--’ to
‘( 3 y)(--y--)’, are, in general, not valid if a description takes the place

” The reasons for this method are explained in detail by Russell in [Denoting]; it has been
applied by Russell and Whitehead in the construction of the system of [P.M.], see I, 66 £f.
and 173 fi.

§ 8. FREGE’S METHOD FOR DESCRIPTIONS 3S

of the individual constant 'a’; here the uniqueness sentence for the de-
scription must be taken as an additional premise. A further disadvantage
of RusseU’s method is the following: A sentence like Q(w) (Pjc) ’ can be
transformed in two ways. Either this whole sentence is taken as 7-2 and
transfocmed into the corresponding sentence of the form 7-4; or the part
‘Q(i 2:) (Pa:)’ is taken as 7-2, transformed into the corresponding sentence
of the form 7-4, and then prefixed again with the sign of negation. The two
resulting sentences are not L-equivalent (in distinction to Frege’s meth-
od) ; hence Russell has to lay down an additional convention, which de-
termines for each case what is to be taken as the context 7-2.

§ 8. Frege’s Method for Descriptions

We adopt for our systems a method proposed by Frege for interpreting indi-
vidual descriptions in cases of nonuniqueness. This method consists in choosing
once for all an individual to be taken as dcscriptum for all such cases.

Method HI. Frege^^ regards it as a defect in the logical structure
of natural languages that in some cases an expression of the grammatical
form ‘ the so-and-so’ is a name’"' of one object while in otk : cases it is not;
in our terminology : that some descriptions have a descriptum but others
not. Therefore, he suggests that the rules of a language system should be
constructed in such a way that every description has a descriptum. This
requires certain conventions which are more or less arbitrary; but this
disadvantage seems small in comparison with the gain in simplicity for the
rules of the system. For instance, specification and existential generaliza-
tion are here valid also for descriptions fat least in extensional contexts) .

Frege’s requirement can be fulfilled in various ways. The choice of a
convenient procedure depends upon the particular features of the lan-
guage system, especially upon the range of values of the variables in
question. There are chiefly two methods which deserve consideration; we
call them Ilia and Illb. We shall explain them and then use Illb for our
systems.

Method Ilia. Frege’® himself constructs a system without type dif-.
ference between individuals and classes; that is to say, he counts both
classes and their elements as objects, i.e., as values of the individual vari-
ables. To any of those descriptions which do not satisfy the condition of
uniqueness he assigns as descriptum the class of those objects which fulfil

•» [Sinn], pp. 39-42.

For the question of English translations for Frege’s terms, see below, p. ii8, n. 21.

^ [Grundgesetze], I, 19.

$6 I. THE METHOD OF EXTENSION AND INTENSION

the scope. Thus different descriptions of this kind may have different
descripta.

Method inb. A simpler procedure consists in selecting, once for all, a
certain entity from the range of values of the variables in question and
assigning it as descriptiun to all descriptions which do not satisfy the con-
dition of uniqueness. This has been done in various ways.

(i) If the individuals of the system are numbers, the number o seems
to be the most natural choice. Frege** has already mentioned this possibil-
ity. It has been applied by Gbdel*’ for his epsilon-operator and by myself**
for the K-operator.

(ii) For variables to whose values the null class A belongs, this class
seems to be the most convenient choice. Such a choice has been made by
Quine,** in whose system there is, as in Frege’s, no type difference between
individuals and classes.

(iii) How can Method Illb be applied to a language system whose
individuals are physical things or events? At first glance, it seems impos-
sible to make here an even moderately natural choice of an individual as
common descriptum for all individual descriptions which do not satisfy
the condition of uniqueness. To select, say, Napoleon would be just as
arbitrary as to select this dust particle on my paper. However, a natural
solution offers itself if we construct the system in such a way that the
spatiotemporal part- whole relation is one of its concepts.** Every indi-
vidual in such a system, that is, every thing or event, corresponds to a
class of space-time points in a system with space-time points as indi-
viduals. Therefore, it is possible, although not customary in the ordinary
language, to count among the things also the null thing, which corresponds
to the null class of space-time points. In the language system of things it is
characterized as that thing which is part of every thing.** Let us take ‘ao’

'* [Sinn], p. 42 n.

” K. Gddel, “Uebcr formal unentscheidbare S&tze der Principia Mathematica und ver-
wandter Systeme”, Monatshefte f. Math. u. Physik, XXXVIII (1931), 173-98.

•* [Syntax], § 7. ” [M.L.], p. 147.

This is, for instance, the case with the following systems; a system for certain biological
concepts by J. H. Woodger {The Axiomatic Method in Biology [1937I; The Technique of Theory
Construction [“International Encyclopedia of Unified Science”, Vol. II, No. s (1939)]); a
calculus of individuals by H. S. Leonard and N. Goodman (“The Calculus of Individuals and
Its Uses”, Journal of Symbolic Logic, V [1940], 45-55); and a general system of logic recently
constructed by R. M. Martin (“A Homogeneous System for Formal Logic”, Journal of Sym-
bolic Logic, VIII [1943], 1-23), where the customary symbol of inclusion and the term ‘in-
clusion’ apparently refer to the part-whole relation among things.

»■ In the system by Martin mentioned in the preceding footnote the null thing is indeed
introduced (see op. cii., p. 3, and D7, p. 9), while in the papw-by Leonard and Goodman
there is an explicit “refusal to postulate a null element” {op. cit., p. 46).

§ 8. FREGE’S METHOD FOR DESCRIPTIONS 37

as the name for the null thing; the other things may be called non-null
things. If a system S includes ao among its individuals, then ao seems a
natural and convenient choice as descriptum for those descriptions which
do not satisfy the uniqueness condition. It is true that this procedure re-
quires certain deviations from the ordinary language for the forms of
sentences in S; but these deviations are smaller than we might expect at
first glance. For most of the universal and existential sentences, the trans-
lation into S is straightforward, i.e., without change in structure; in other
cases ‘non-nuU’ must be inserted. [Examples: The sentence ‘There is no
thing which is identical with the king of France in 1905’ is translated into
a sentence of S of the form ‘There is no non-null thing . . .’. On the other
hand, no such change in form is necessary for the sentence ‘All men are
mortal’ and not even for ‘ There is no man who is identical with the king of
France in 1905’, because it follows from any suitably framed definition for
‘man’ that every man is a non-null thing.]

In our further discussions we assume for our system S, that Frege’s
Method mb is applied and that the individual constant ‘a*’ is used
for the common descriptum of all descriptions which do not satisfy the
uniqueness condition. We leave it open which individual is meant by
‘a’'"’; it may be the null thing a#, if this belongs to the individuals in S,;
it may be o, if numbers belong to the individuals (as, for instance, in S,) ,
but it may as well be any other individual. Consequently, a sentence con-
taining a description is now interpreted in a way different from Russell’s.
The translation of 7-2 into M is now as follows (instead of 7-3) ;

8 - 1 . ‘ Either there is an individual y such that y is the only individual for
which . . y . . holds, and --y--; or there is no such individual, and
- - a* - [In the previous example: ‘Either there is an individual y such
that y is the only author of Waverley, and y is human; or there is no such
individual y (that is to say, there is either no author or several authors
of Waverley), and a* is human’.]

Hence, the sentence 7-2 containing the description is L-equi valent in S,
to the following (instead of to 7-4) :

8 -2. ‘(Sy) [(»:)(. . a; . . = (a; = y)) • - - y - -] V [~(ay)(a:)(. .x..=
{x = y)) » - - - -] ’. (In the example: ‘ ( 3 y) [(x) (Axw = (x = y))

• Hy] V ( 3 y)(a:) (Aarw = (x = y)) • Ha*]’.)

Here again, as in the case of Russell’s method, we may set up either a
contextual definition for 7-2 with 8-2 as definiens, or semantical rules for
the iota-operator as a primitive sign such that 7-2 becomes L-equivalent
to 8-2.

38

I. THE METHOD OF EXTENSION AND INTENSION

The accompanying table gives a, survey of the various methods just
explained for dealing with descriptions in the case of nonuniqueness. The
case of uniqueness is not represented because its treatment is the same
with all authors.

INTERPRETATION OF DESCRIPTIONS IN THE CASE OF
NONUNIQUENESS

HiLBSKT'

Beknavs

Russell

Fkege I

Quine

System

or

l^os

System

Sg

(a)

(b)

Method 1

Method II

Method

Ilia

Method

Illb

Method

Illb

Method

Illb

Method

nib

Description
is mean-
ingless

No descriptum;
the sentence is
meaningful
but false

■f(.

0

A

Null

thing

&1>

a*

Some brief remarks may be made on descriptions with variables of other
than individual type, especially predicator variables, functor variables,
and sentential variables. (This is a digression from the study of our sys-
tems Si, etc., which contain only individual variables.) Here it is easy to
make a natural choice of a value of the variable as a descriptum for those
descriptions which do not satisfy the condition of uniqueness. If an indi-
vidual has been chosen as a* (it may be ao or o or anything else), then we
might call one entity in eveiy type the null entity of that type, in the
following way; In the type of indivi duals it would be a*; in any predica-
tor type, the null class or null relation of that type, e.g., for level one and
degree on e the null class A ; in the type of propositions, tKeTT-TaTse pTo'posi-
tion; in any type” of Tuhctio'ns, that function which has as value IdFair
argumentsTSie hull entity of the type in question. Then we may take as
descriptum in the case of nonuniqueness the null entity of the type of the
description variable.

For the sake of simplicity, the following explanations are restricted to
extensional systems. Let ‘/’ and ‘g be predicator variables of level one and
degree one. Let ‘- - (i/)(. . / . indicate, in analogy to 7-2, a sentence
containing a description of the type of ‘/’, hence a description for a class
or property. This sentence is L-equivalent to the following, in analogy
to 8-2:

‘(3^)[a)(../.. - (/-f)) .--^--1 V h (3g)(/) (../.. = (/ = g))

--A--]’.

§ 9. INTENSIONS OF INDIVIDUAL EXPRESSIONS

39

The uniqueness condition here occurring says that there is a property g,
such that for those / and only those, which are equivalent to g, .. f ;
in other words, there is exactly one class g such that . . g ■ .V^ence
here the uniqueness applies to extensions, not to intensions. This is in
analogy to 7-1 and 7-3; for, as we shall see later, the extensions of indi-
vidual expressions are individuals.

However, if the system contains lambda-operators for the formation of
predicators, then descriptions with predicator variables are not necessary,
they can be replaced by lambda-expressions. In this case we can transform
not only a sentence containing the description as in the earlier case but
the description itself into an L-equivalent expression. The description
‘ •/■ •)’ is L-equivalent to the lambda-predicator ‘ (Xa;) [( 3 g) ((/)

[../.. = {/ = g)] -gx)]’.

In a similar way, for every description of a function (containing an
iota-operator with a functor variable) there is an L-equivalent functor
formed with a lambda-operator. And for every description containing an
iota-operator with a sentential variable there is an L-equivalent sentence
without an iota-operator; however, in an extensional system these de-
scriptions with sentential variables are rather useless anyway.

In view of these results, it seems convenient in the primitive notation
of a system (at least in an extensional one) to use the iota-operator, if at
all, for individual descriptions only, and then to use the lambda-operator
for the formation of predicators and functors.^*

§ 9 . Extensions and Intensions of Individual Expressions

It is found to be in accord with our earlier conventions, to take as the exten-
sion of an individual expression the individual to which it refers. The intension
of an individual expression is a concept of a new kind; it is called an individual
concept.

Let us consider some examples of F-equivalehce and L-equivalence of
individual expressions. We assume the following as a historical fact:

9 - 1 . Assumption. There is one and only one individual which is an author
of Waverley, and this individual is the same as Walter Scott.

Then the descriptum of ‘ (ix) (Axw)’ is that individual which is author of
Waverley and not a* and ‘ (ix) (Axw) = s’ is, according to the rule 3-3,
true, but not L-true; hence it is F-true. This leads to the following result,
according to the definitions 3-5:

3 * Several forms of systems with predicators and functors built with lambda-operators
have been constructed by Church, see especially The Calculi of Lambda-Conversion (“Ann. of
Math. Studies”, No. 6 [1941]).

40

I. THE METHOD OF EXTENSION AND INTENSION

9 - 2 . ‘(w)(Aj;w)’ is equivalent to ‘s’, but not L-equi valent, hence F-
equivalent.

On the other hand, let us compare the two descriptions' (la:) (Ha: • Aarw) ’
and ‘ (la;) (RAa: • Axw) Let us see what we can find out about them if we
make use of the rules of S,, especially 1-2, but not of any historical or other
factual knowledge. If there is exactly one individual which is both human
— or, which means the same, a rational animal — and an author of Waver-
ley, then the descriptum of each of the two descriptions is this individual;
otherwise the descriptum of each is a*. Thus, in either case, the de-
scriptum of the first description is the same individual as that of the
second. Hence, according to rule 3-3, the sentence ‘(i2:)(Ha: • Ajcw) =
(w)(RA2; • Axw)’ is true; it is, moreover, L-true because we have shown
its truth by using merely the semantical rules. Therefore, the two descrip-
tions are L-equivalent.

We found earlier that individual expressions are equivalent if and only
if they are expressions for the same individual (3-12). Hence, according
to the definition of identity of extensions C5-1), individual expressions
have the same extension if and only if they are expressions for the same
individual. Therefore, it seems natural to regard as extensions of indi-
vidual expressions the individuals themselves:

9 - 3 . The extension of an individual expression is the individual to which
it refers, (hence, the .descriptum, if it is a description).

Since we adopted Frege’s method, every description has exactly one
descriptum. Hence, on the basis of the convention just made, there is no
ambiguity with respect to the extension of an individual expression. For
instance, the extension of ‘s’ is the individual Walter Scott, and the same
holds for each of the three descriptions discussed above as examples. If
there were none or several authors of Waverley, then the extension of
‘ (u) (Axw) ’ would be the individual a*.

Now let us look for entities which we might regard as intensions of
individual expressions. According to our definition for the identity of in-
tensions (5-2), the intension must be something that L-equivalent indi-
vidual expressions (for example, the two descriptions above containing
‘H’ and ‘RA’) have in common. We have earlier found entities which
seemed suitable as intensions of designators of other types; for sentences,
propositions; for predicators, properties or relations; for functors, func-
tions. Thus, in these cases, the intensions a re those €nlitje5i ,yr.lfich are
sgge.tiines. regaakti-as .the laejinings pf the expressions in questio n; and,
Jn_the-cas&«f predicat£US_and^^ctors, the intensions are concepts of cer-

41

§ 9. INTENSIONS OF INDIVIDUAL EXPRESSIONS

tain types. Now it seems to me a natural procedure, in the case of indi-
"vidual expressions, likewise to sneak of concents, hut of concents of a par-
ticular tvp p| namply tViP mHiiriHiial i-ypp Although it is not altogether
customary to speak here of concepts in this sense, still it does not seem to
deviate too much from ordinary usage. I propose to use the term ^indi-
vidual concept' fo r this type of concept. Thus we say;

9-4. The intension of an individual expression is the individual concep t
expressed hv iL

Examples:

9-6. The intension of ‘s’ is the individual concept Walter Scott.

9-6. The intension of ‘(ix)(Aa:w)’ is the individual concept The Author
Of Waverley.

(Here, and further on, in translating descriptions into M, we omit for
brevity the phrase ‘or if there is not exactly one such individual’.) In-
stead of saying in the customary but ambiguous terminology that the two
L-equivalent descriptions discussed above have the same meaning, we say
now that they have the same intension and that their common intension
is the individual concept The Human Author Of Waverley, which is the
same as the individual concept The Rational Animal Author Of Waverley.
On the other hand, the following are three different individual concepts:
the one just mentioned, the individual concept Walter Scott, and the indi-
vidual concept The Author Of Waverley. Here again the intensions of given
expressions, and the identity or nonidentity of these intensions, can be
determined on the basis of the semantical rules alone.

We have seen earlier how a sentence containing a predicator can be
translated into M, that is, English, in different ways. Thus, for the sen-
tence ‘Hs’, we had, in addition to the simple translation ‘ Scott is human’,
two more explicit translations, one of which used the term ‘property’ and
the other the term 'class’ (see 4-2 and 4-3). In these two explicit transla-
tions, ‘s’ was still simply translated by ‘Scott’. Now, however, we have
seen that, corresponding to the distinction between classes and properties,
we have in the case of individual expressions the distinction bdrween indi-
viduals and individual concepts. Hence, we may use in M instead of
‘ Scott’ the more explicit phrases ‘ the individual Scott’ and ‘ the individual
concept Scott’. Since the distinction is perhaps clearer for a description
than for an individual constant, let us take, instead of ‘ Hs’, the sentence
‘H(w)(Acw)’. In addition to the simple translation ‘the author of
Waverley is human’, we have here four more explicit translations in which
both to ‘The Author Of Waverley’ and to ‘Human’ a characterizing word

4 »

I. THE METHOD OF EXTENSION AND INTENSION

is added. Two of these translations are pure, two mixed. Of the two pure
translations, the first contains two references to extensions, and the second
two references to intensions; these translations are as follows:

‘The individual The Author Of Waverley belongs to the class Human’.

‘The individual concept The Author Of Waverley is subsumable under
the property Human’.

'Since it is not customary to speak about individual concepts, there is no
word in customary usage for the relation between an individual^ncept
and a property'corresponding to the ^ment-relation between indi-
vidual and\ class; we have used here for this relation the word ‘subsum-
able’ (in the sense of ‘ truly subsumable’), but we shall not use it further
on. Of the two mixed translations, which contain a reference to an exten-
sion and a reference to an intension, we shall give at least one, because it is
not too far from customary usage:

‘The individual The Author Of Waverley has the property Human’.

Thus we find here a multiplicity of possible translations into M, some of
them rather cumbersome and strange-looking. This multiplicity seems in-
evitable as long as we wish to distinguish explicitly between classes and
properties and between individuals and individual expressions. The prob-
lem of whether and by which means this apparent multiplicity of entities
and the corresponding multiplicity of formulations can be reduced will be
discussed later (§§ 33 f.).

§ 10. Variables

We found earlier that the extension of a predicator ‘P’ is a class, and its in-
tension is a property. Therefore, a variable of the same type (e.g., ‘/’) refers
both to classes and to properties; we say that classes are its value extensions, and
properties its vahie intensions. Analogously, for a variable of the type of sen-
tences (e.g., ‘p’), the value extensions are truth-values, and the value inten-
sions are propositions. Finally, the value extensions of an individual variable
(e.g., ‘x’) are individuals, and its value intensions are individual concepts.

Quine has repeatedly prointed out the important fact that, if we wish to
find out what kind of entities somebody recognizes, we have to look more at
the variables he uses than at the constants and closed expressions. “The
ontology to which one’s use of language commits him comprises simply the
objects that he treats as falling . , . within the range of values of his vari-
ables.”^^ I am essentially in agreement with this view, as I shall presently
explain. But, first, I wish to indicate a doubt concerning Quine’s formula-
tion; I am not quite clear whether the point raised is not perhaps of a
» [Notes], p. 118; see also his [Designation].

§ 10 . VARIABLES

merely terminological nature. I should prefer not t^o use the ‘n*anLtg^’
f or the remgnitinn p f tJip admission nf This USe seems

to me to be at least misleading; it might be understood as implying thaL
the decision to use cer t ain 1r i »d« - of v a ri a hles musti lfc
metaphysical convictions. In my view, however, the choice of a certain
language structure and, in particular, the decision to use certain type s of .
variables is a practical decision like the an Tpgipimpn^t ; it de-

pends chiefly upon the purposes for which the instrument — here the lan-
guage — is intended to be used and upon the properties of the instrument.

I admit that the choice of a language suitable for the purposes of physics
and mathematics involves problems quite different from those involved in
the choice of a suitable motor for a freight airplane ; but, in a sense, both
are engineering problems, and I fail to see why metaphysics should enter
into the first any more than into the second. Furthermore, I, like many
other empiricists, regard the alleged questions and answers occurring in
the traditional realism-nominalism controversy, concerning the onto-
logical reality of universals or any other kind of entities, as pseudo-ques-
tions and pseudo-statements devoid of cognitive meaning. I agree, of
course, with Quine that the problem of “Nominalism” as he interprets
it’^ is a meaningful problem: it is the Question of whether all na^ral sci-
ence can be expressed in a “npminalistic” language, that is. one contain-
i ng only individual variables whose values are concrete obfects. not
classes, proper ties, and the like . However, I am doubtful whether it is
advisable to transfer to this new problem in logic or semantics the label
‘nominalism’ which stems from an old metaphysical problem.

The sense in which I agree with Quine’s thesis that “to be is to be the
value of a variable” will become clear by the following example: Suppose
somebody constructs a language not only as a subject matter of theoretical
investigations but for the purpose of communication. Suppose, further,
that he decides to use in this language variables ‘m’, ‘n’, etc., for which
all (natural) numerical expressions (e.g., ‘o’, ‘3’, ‘2 -f- 3’, etc.) and only
those are substitutable. We see from this decision that he recognizes
natural numbers in this sense: he is willing to speak not only about par-
ticular numbers (e.g., ‘ 7 is a prime number’) but also — and this is the de-
cisive point — about numbers in general. He will, for example, make state-
ments like: ‘for every m and n, m + n = n + m’ and ‘there is an m
between 7 and 13 which is prime’. The latter sentence speaks of the exist-
ence of a prime number. However, the concept of existence here has
nothing to do with the ontological concept of existence or reality. The sen-

34 [Designation], p. 708.

44

I. THE METHOD OF EXTENSION AND INTENSION

tence mentioned means just the same as ‘it is not the case that for every
m between 7 and 13 , w is not prime’. By the same token, we see, further-
more, that the user of the language is willing to recognize the concept
Niunber. Generally speaking, if a language (of ordinary structure) con-
tains certain variables, then we can define in it a designator for the range
of values of those variables. In the present case, the definition is: “ ‘Num-
ber’ for‘(Xw)(w = w)’ ” or, if the language in question does not con-
tain abstraction operators, “ ‘Number(»t)’ for ‘m = m’ ”. [In the de-
finiens, any matrix ‘. . w . may be used which is L-universal, that is,
such that ‘ (m) (. . . .) ’ is L-true.] It is important to emphasiz e tl^

point just made that, once you admit certain variables, vou are bound
to admit the corresponding universal concept. It s^riis to me that some
philosophers (not Quine) overlook this fact; they do not hesitate to admit
into the language of science variables of the customary kinds, like senten-
tial variables ('/)’, ‘ q', etc.), numerical variables, perhaps also predicator
variables at least of level one, and other kinds; at the same time, however,
they feel strong misgivings against words like ‘proposition’, ‘number’,
‘property’ (or ‘class’), ‘function’, etc., because they suspect in these
words the danger of an absolutist metaphysics. In my view, however, the
accusation of an absolutist metaphysics or of illegitimate hypostatizations
with respect to a certain kind of entities, say propositions, cannot be made
against an author, merely on the basis of the fact that he uses variables of
the type in question (e.g., ‘ p', etc.) and the corresponding universal word
(‘ proposition’) ; it must be based, instead, on an analysis of the statements
or pseudo-statements which he makes with the help of those signs.

Quine’s thesis and my remarks in connection with it concern the lan-
guage which somebody not only analyzes but uses, hence, with respect to
semantical discussions, the metalanguage. Now let us look at the role of
variables in an object language 5. If S is given, then a metalanguage M
intended for the semantical analysis of S must be rich enough in relation
to 5. In particular, M must contain variables whose ranges of values cover
those of all variables in 5 (and, as Tarski has shown, even go beyond this
in order to make possible the definition of ‘ true in S’). Let us further pre-
suppose here, as in the previous discussions, that M enables us to speak in
general terms about the extensions and intensions of predicators, sen-
tences, and individual expressions of 5.

Let 5 (in distinction to S,) contain not only individual variables but also
those of other types. Let us begin with variables ‘/’, ‘g’, etc., of the type
of predicators of level one and degree one. With respect to a predicator,
say ‘H’ in S„ we have distinguished between its extension, the class

§10. VARIABLES

4S

Human, and its intension, the property Human. A sentence ‘. . H .
containing ‘H’ can be translated into M in different ways; we may use
either the word ‘ human’ alone or the phrase ‘ the class Human’ or ‘ the
property Human’ (see, as an example, the translations of ‘Hs’ in § 4) ; we
have seen that this involves merely a difference in formulation. Now in
S, we can deduce from '. . H . .’ the existential sentence '(3/} (• • / • •)’•
For the translation of this sentence into M we have again three forms,
corresponding to the three forms mentioned for the transla tion of
‘. . H . .’;

(i) ‘There is an / such that . . / . .’,

(ii) ‘There is a class / such that . . / . .’,

(iii) ‘There is a property / such that

As ‘H’ is an expression both for the class Human and for the property
Human, ‘/’ is thus a variable both for classes and for properties. Since we
regarded the class Human as the extension of ‘H’, we shall now regard it
as one of the value extensions of ‘/’; and, analogously, we take the
property Human as one of the value intensions of ‘/’. Let us ca ll &e
cl osed expressions substitutable for a certain variable of any kind^tiie
value expressions of thsttj/ariaWe. Then the following holds generally,
for variables of any kind .

10-1. The extension of a value expression of a variable is one of the value
extensions of that variable.

10-2. The intension of a value expression of a variable is one of the value
intensions of that variable.

For variables of the type of sentences, say ‘ p’, ‘q’, etc., the situation is
analogous. Their value extensions are truth-values; their value intensions,
propositions. Let‘ . . Hs . .’ be a sentence containing ‘Hs’ as a proper sub-
sentence. We may translate ‘ . . Hs . .’ into M in various ways. One pos-
sible translation contains simply the phrase ‘(that) Scott is human’. Of
the two more explicit translations, one contains the phrase ‘the truth-
value that Scott is human’, and the other ‘the proposition that Scott is
human’, in accord with our earlier results concerning the extension and the
intension of ‘Hs’ (6-3 and 6-4). Now in S, we may infer from' . . Hs . .’ the
existential sentence ‘(3/))(. . p . Corresponding to the three transla-
tions of ‘. . Hs . .’, we have three translations of this existential sentence:

(i) ‘There is a such that . . p .

(ii) ‘There is a truth- value p such that . . p .

(iii) ‘There is a proposition p such that . . p . .’.

46 I. THE METHOD OF EXTENSION AND INTENSION

The treatment of individual variables is not essentially different from
that of the other kinds of variables. But, owing to the unfamiliarity of
individual concepts, our conception here may seem less natural at first
glance. We considered earlier the sentence ‘H:(ia;)(Aa:w)’ containing a de-
scription. In addition to the simple translation ‘ the author of Waverley is
human’, we had several more explicit translations containing the phrases
‘ the individual’ and ‘ the individual concept’ (at the end of the preceding
section). From the sentence with the description (or from the simpler
sentence ‘Hs’) we may deduce ‘(3*) (Ha:)’. Corresponding to the earlier
translations of the former sentence, we have the following translations of
this existential sentence:

(i) ‘There is an * such that x is human’.

(ii) ‘There is an individual x such that x belongs to the class Human’.

(iii) ‘There is an individual concept x such that x is subsumable under
the property Human’.

(iv) ‘There is an individual x such that x has the property Human’.

Thus the value extensions of individual variables are individuals, their
value intensions are individual concepts. The multiplicity of the formula-
tions and the strangeness of some of them are the same here as in the
preceding section. Our later attempt at a simplification will apply to the
present situation, too.

§ 11. Extensional and Intensional Contexts

An expression occurring within a sentence is said to be interchangeable with
another expression if the truth-value of the sentence remains unchanged when
the first expression is replaced by the second. If, moreover, the intension of the
sentence remains unchanged, the two expressions are said to be L-inlerchange-
able. We say that a sentence is extensional with respect to an expression oc-
curring in it or that the expression occurs in the sentence within an extensional
context, if the expression is interchangeable at this place with every other ex-
pression equivalent to it. We say that the sentence is intensional with respect to
the expression, or that the expression occurs within an intensional context, if
the context is not extensional and the expression is L-interchangeable at this
place with every other expression L-equivalent to it. (The definitions actually
given in this section are wider than here indicated; they refer not only to
sentences but to designators of any type.) It is found, in accordance with custom-
ary conceptions, that all sentences of the system S„ which contains only the
ordinary connectives and quantifiers but no modal signs, are extensional and
that a sentence in Sa of the form ‘N(. . .)’, where ‘N’ is a sign for logical neces-
sity, is intensional.

Suppose that we replace an expression (designator or not) which occurs
within a designator by another expression. It may happen that the exten-

§ 11. EXTENSIONAL AND INTENSIONAL CONTEXTS 47

sion of the designator is not thereby changed; in this case we call the two
expressions interchangeable within the designator. If, moreover, the in-
tension of the designator remains unchanged, we say that the two expres-
sions are L-interchangeable within the designator. The subsequent
definitions for these concepts in technical terms (ii-ia) refer not to ex-
tension and intension but, instead, to equivalence and L-equivalence. Two
further concepts are defined (ii-ib), which apply to the case in which the
conditions mentioned are fulfilled for all sentences. The system 5” to which
these and the later definitions (11-2) refer may be one of our systems S„
S3, S3, or a similar system with the same types of designators; it is sup-
posed that 5 contains descriptive predicates, and hence factual sen-
tences,^^ and also individual descriptions with those predicates. [Thus S
may be PM', but not PM, in § 26; it may be ML', but not ML, in § 25.]
5 may, in distinction to our systems, also contain variables for the non-
individual types of designators.

11-1. Definitions

a. An occurrence of the expression 21, within the expression 2I< is (,1)
interchangeable, (2) L-interchangeable with 21/ (in S) =Df 2Ii is
a designator and is (i) equivalent, (2) L-equivalent to the expression
21/ constructed out of 2f, by replacing the occurrence of 21, in
question by 21/.

b. 21,- is (i) interchangeable, (2) L-interchangeable with 21/ in
the system S =ot any occurrence of 21,- within any sentence of S' is
(i) interchangeable, { 2 ) L-interchangeable with 21/.

Consider a particular occurrence of a designator 21,- within a designator
2li. The situation may be such that the extension of 21. depends merely
upon the extension of 21,, that is to say, it remains unchanged if 21, is
replaced by any other expression with the same extensio n. In this case we
shall sa y that 21, is extensi on al with respect to that occurrence of
(i i^^^aJT'We must here refer to a particular occurrence; for, if iTi contains

js The fact that a restriction of this kind is necessary was pointed out to me by Alonzo
Church. If 5 is a system of modal logic which, like Lewis’ system of strict implication, con-
tains no descriptive predicates and hence no factual sentences, then any two equivalent sen-
tences are L-equivalent and hence are L-interchangeable even within a modal sentence of the
form ‘N(. . Thus the latter sentence would fulfil the condition of extensionality as stated
below in ii-ab; in fact, however, a modal sentence is, of course, to be regarded as intensional
in the customary sense. To state definitions of ‘extensional’ and ‘intensional’ which are ap-
plicable also to systems containing only L-determinate sentences or no closed sentences at all,
it would be necessary to refer not only to closed designators occurring as parts but also to the
values of the designator variables and to the corresponding values of propositional functions
expressed by matrices (for example, to the values of ‘p’ and the corresponding values of
‘Np’). In order to avoid this complication in our present discussion, we restrict the systems
S as indicated in the text.

48 I. THE METHOD OF EXTENSION AND INTENSION

several occurrences of ^ it may happen that one occurrence fulfils the
above condit ion, while another does not. ll tEe condition is fulfille37w^
shall ako saj sometimes ^at occurs within at the place in que stion^
in an extensional context.

11-2. Definitions

a. The ext>ressi o n jli is extensional with resPect to a certa in occur -
rence of 81) within Sli (in the system S) = Df 2t. and 21, are desjgna^tors;
the occurrence fh“ question of 31, within SSi is interchangeable with
any expression equivalent to 8 / (in S).
h* The expression %i.h.ext£nsiQnaL (in is a de signator (in

S ) ; 2t< is extensional with respect to any occurrence of a desig nator

within 2ii (in 5). ^

c. The semantical system S is extensional =Df every sentence 3 . 5
is extensional.

If the condition in ii-aa or b or c is not fulfilled, we shall use the term
‘ nonextensionaf^he term* iniensionaV (u- 3 ) will be used not, as is some-
times done, as synonymous with’tvonextensional’, but in a narrower sense,
namely, in those cases in which the condition of^xtensionality is not ful-
filled out the analogous condition with respect to intension is fulfilled.
The latter condition means that the intension of the whole remains un-
changed if the subexpression is replaced by one with the same intension ;
the technical definition ( 11 - 3 ) does not refer to intension but uses, in-
stead, the concepts of L-equivalence and L-interchangeability.

11-3. Definitions

^a. The expression 21, is intensional with respect to a certain occur-
rence of 21, within 21, (in S) =Df 2(< and 21, are designators; 21, is not
extensional with respect to the occurrence in question of 21 ,- within
21 ,; this occurrence of 21 ,- within 2 f, is L-interchangeable with any
expression L-equivalent to 21, (in S).

^ b. The expression 2li is intensional (in S) =di 21, is a designator; 21,
is, with respect to any occurrence of a designator within 21 ,-, either
extensional or intensional, and is intensional with respect to at least
one occurrence of a designator.

\yc. The semantical system S is intensional =Df every sentence in S
is either extensional or intensional, and at least one is intensional.

We shall sometimes call a sentential connective or a predicator con-
stant extensional, if every full sentence of it is extensional with respect to
the argument expressions; and we shall use the term ‘ intensional’ anal-
ogously.

1 11. EXTENSIONAL AND INTENSIONAL CONTEXTS 49

Note that the terms ‘extension’ and ‘intension’ occur only in the in-
formal explanations and not in the definitions ii-i, 11-2, and 11-3 them-
selves. Thus these definitions do not presuppose any problematic entities.
They use, instead, the terms ‘equivalent’ and ‘L-equivalent’, which, as
mentioned earlier (§5), are unproblematic and can be defined in an exact
way.

The terms ‘ interchangeable’, ‘L-interchangeable’, ‘ extensional’, and ‘ in-
tensional’ have been defined here in a general way so that the whole ex-
pression ?I, may be a designator of any of the types occurring in our sys-
tems.'^^hese terms find their most important application, however, in
those cases in which 21. is a sentence; and in our further discussions we
shall use them chiefly for cases of this kind.

The concepts just defined will become clearer with some examples. The
whole expression 21 , is a sentence in all these examples. The subexpression,
21 y, is, in the first three examples, a sentence; in the later examples a
designator of another type.

Example I. A sentence ‘. . V - is extensional with respect to either
of its components. And, generally, as is well known, any full sentence of
the ordinary connectives, ‘ V’, ‘ D ’, and ' = is extensional with

respect to its (immediate) component or componentsVThese connectives
and the connections for which they stand are, indeed, often called exten-
sional;’* following Russell, the connections are usually called truth-
functions.

Example II. Anticipating later explanations (chap, v), let us use here
the system S3, which contains the signs of S, and, in addition, ‘N’ as a
modal sign for logical necessity in such a way that, if ‘ . . .’is any L-true
sentence, ‘N(. . .)’ is true and, moreover, L-true; and if ‘. . .’is any sen-
tence not L-true, then ‘N(. . .)’ is false and moreover L-false (see 39-3).
Let ‘C’ be an abbreviation for an F-true sentence (e.g., for ‘Hs’); then
‘ C’ is true but not L-true. As is well known (see the example following
*-2) ‘C V~ C’ is L-true. Hence:

11 - 4 . ‘C’ and ‘C V~C’ are equivalent but not L-equivalent.

According to the given explanations for ‘ N’, we have :

11 - 6 . ‘N(C V~C)’ is true and, moreover, L-true.

On the other hand, since ‘C’ is not L-true, ‘N(C)’ is false. Therefore,
‘N(C V~C)’ and ‘N(C)’ are not equivalent. It follows, according to the
definition ii-ia, that the occurrence of ‘C’ within ‘N(C)’ is not inter-

The concept of extensionality of connections and connectives and the corresponding
concept of L-eztensionality are discussed in more detail in [II], § 12.

I. THE METHOD OF EXTENSION AND INTENSION

SO

changeable with ‘C V~C’. This, together with 11-4 and the definition
ii-2a, leads to the following result;

11 - 6 . ‘N(C)’ is nonextensional with respect to ‘C’.

This result is well known; generally, full s entences of modal sig ns are
nonextensional with respect to thei r co mponents.: in custQisa rv terms,
moffalTfies areTiot truth-functions.” The same consideration shows that
the occurrence bT‘C’V~t!* within ‘N(C V~C)’ is not interchangeable
with ‘C’. Thus we obtain (again with 11-4) :

11 - 7 . ‘N(CV~C)’ is nonextensional with respect to the subsentence
‘C V-^C’.

Further, let ‘D’ be any sentence L-equivalent to ‘C V~C’. Then ‘D’
is likewise L-true; and hence ‘N(D)’, too. We found that ‘N(C V~C)’
is L-true (11-5). Since any two L-true sentences hold in the same state-
descriptions (2-2), they are L-equivalent to each other (2-6). Thus
‘N(CV~C)’ and ‘N(D)’ are L-equivalent. Therefore, according to the
definition ii-ia, the occurrence of ‘C V~C’ within ‘N(C V~C)’ is L-
interchangeable with any sentence which is L-equivalent to ‘CV~C’.
This, together with 11-7 and the definition ii-3a, yields:

11 - 8 . ‘N(C V ~C)’ is intensimql with resii£.ct . .to ..the. . sub sent ence
‘CV'^C’.

Example III. The sentence ‘Hs’ is true in S,; it remains true if ‘H’ is
replaced by any equivalent predicator, for instance, by and like-

wise if ‘s’ is replaced by any equivalent individual expression, for instance,
by the description ‘ (u) (A;cw)’ (9-2). Therefore:

The results ii-6 and 11-7 refute Church’s opinion that (on a certain assumption, see
below) “Carnap’s definition of ‘extensional’ fails in that under it every language (every
semantical system) is extensional, even those which contain names of propositions and
modal operators” ([Review C ], p 504). The definition of ‘extensional’ here referred to is
[I], Dio-20 and Dio- 21, p 43; it is essentially the same as ii-i and 11-2 in the present
section; however, the restriction to systems with factual sentences was omitted. Church is
right in criticizing this omission (see n, 35). However, if the definition is applied to systems
also containing factual sentences, like the example systems in my earlier book [I] and in the
present book, then the definition seems to me to be adequate; at any rate, the examples here
mentioned show that it is certainly not the case that under this definition (either in the
earlier or in the present formulation) all sentences and all semantical systems fulfill the defining
condition for extensionality. Church qualifies his statement by the following condition; “if
the designatum of a sentence is always a truth-value.” [Here the term ‘designatum’, as
Church’s preceding explanations show, is meant in the sense in which I shall use the term
‘nominalum’ in this book (§ 24); this sense is different from that in which I have used the
term ‘designatum’ in [I], see below, § 37). However, this qualification does not change the
situation. Any assumption as to what are the designata (nominata) of sentences is irrelevant
to the question of whether the examples stated in ii-6, 11-7, and 13-4 are extensional or not
on the basis of my definition, because in this definition the concept of the designatum (nomi-
natum) of a sentence is not used.

§ 12. THE PRINCIPLES OF INTERCHANGEABILITY Si

11-9. ‘Hs’ is extensional with respect to both ‘H’ and ‘s’.

Example IV. It can easily be shown that every sentence in S, con-
structed out of predicator constants (like those mentioned in rule 1-2),
individual constants (like those mentioned in i-i), connectives of the kind
mentioned in Example I, universal and existential quantifiers, and iota-
and lambda-operators is extensional with respect to any designators con-
tained in it and hence is extensional (ii-2b). Si is intended to contain
only sentences constructed in this way. Therefore, according to definition
ii-2c:

11- 10. Si is an extensional system.

§ 12. The Principles of Interchangeability

Some theorems are stated concerning interchangeability and L-interchange-
ability in extensional and intensional contexts.

The following theorems, which we call principles of interchangeability,
follow from our previous definitions of interchangeability and L-inter-
changeability (n-i), extensionality (11-2), and intensionality (11-3). The
system S, to which the theorems of this section refer, is supposed to be
either one of our systems S,, Sj, Sj, or a similar system as specified earlier
(see the explanation preceding ii-i).

12- 1. First Principle of Interchangeability. Let be a sen-

tence (in the system S) which is extensional with respect to a certain
occurrence of the designator 2(„ and the corresponding sentence

with an occurrence of Sit instead of that of Sly; analogously for ‘. . m . .’
and ‘ . V . in c.

a. If Sly and SI* are equivalent (in S), then the occurrence in question
of Sly within . . SI, . . is interchangeable with Sit (in S).

b. (Sly ^ Sit) D (. . Sly . . s . . SI* . .) is true (in S).

c. Suppose that 5 contains variables for which 31, and 31* are substitut-
able, say ‘ m’ and‘z»’; then '(«)(»)[(« ^ v) D (. . u . . = . . v . .)]’ is
true (in S).

Statement 12-ia follows immediately from the definition ii-2a; and b
and c follow from a by the general definition of equivalence (3-5a). The
forms b and c have the advantage that here the principle is represented
by a sentence in the object language 5 itself. The form c requires suitable
variables. In the system S,, for instance, form c is applicable only with
individual variables and hence states only the interchangeability of indi-
vidual expressions, while forms a and b apply also to predicators and
sentences in S,.

s?

I. THE method of extension AND INTENSION

12-2. Second Principle of Interchangeability. Let . . 8Iy . . be a sen-
tence (in S) which is either extensional or intensional with respect to
a certain occurrence of the designator 31 /, and . . SI* . . the corresponding
sentence with ?(*.

a. If Hyand 21 * are L-equivalent (in S), then the occurrence in question
of 2 Iy within is L-interchangeable and hence interchangeable

with 2 f* (in S).

Formulations b and c of the second principle analogous to 12-ib and c are
possible only with the help of a modal sign, hence only with respect to a
nonextensional language system. They will be given later (39-7 b and c).

The following theorems follow from the two principles just stated, with
the help of the definitions of extensional and intensional systems (rr-2c
and 11-3C):

12-3. Let S be an extensional system (for instance, S,, see Example IV in

§ ”)•

a. Equivalent expressions are interchangeable in S.

b. L-equivalent expressions are L-interchangeable in 5 .

Examples, a. Equivalence and therefore interchangeability in Si hold
for the following pairs of expressions; (i) ‘H’ and ‘F*B’ (see 3-8); (ii)
‘Hs’ and' (F»B)(s)’; (iii) ‘s’ and ‘(iac)(A2:w)’ (see 9-2). b. L-equivalence
and therefore L-interchangeability in S, hold for the following pairs of
expressions: (i) ‘H’ and ‘RA’ (see 3-11); (ii) ‘Hs’ and ‘RAs’; (iii) ‘(ix)
(Hx.Axw)’ and ‘(ix)(RAx«Axw)’ (see § 9).

12-4. Let 5 be an intensional system (for instance, with the modal sign
‘N’, see Example II in § ii and § 39).

a. Equivalent expressions are interchangeable in 5 , except where they
occur in an intensional context (for example, in the system Si.-
except in a context of the form ‘N(. . .)’).

b. L-equivalent expressions are L-interchangeable in 5 .

Examples for S3, a. Let'C’ be F-true, as in Example II, §11. Then ‘C’
and ‘CV~C’ are equivalent (see 11-4). The sentence ‘(CV^C)*
N(C V~C)’ is true (see 11-5). Within this sentence the first occurrence
of ‘C V~C’ is interchangeable with'C’, while the second is not. b. For
the pairs of L-equivalent expressions in Si mentioned above, L-equiva-
lence in S3 and therefore L-interchangeability in S3 likewise hold.

§ 13. SENTENCES ABOUT BELIEFS

S3

§ 13. Sentences about Beliefs

We study sentences of the form ‘John believes that . . If here the sub-
sentence is replaced by another sentence L-equivalent to it, then it may
be that the whole sentence changes its truth-value. Therefore, the whole belief-
sentence is neither extensional nor intensional with respect to the subsentence
‘. . .’. Consequently, an interpretation of belief-sentences as referring either to
sentences or to propositions is not quite satisfactory. For a more adequate
interpretation we need a relation between sentences which is still stronger than
L-equivalence. Such a relation will be defined in the next section.

We found that ‘ ... V ’is extensional with respect to the subsen-

tence indicated by dots, and that' N(. . .) ’ is intensional. Can there be a con-
text which is neither extensional nor intensional? This would be the case if
(but not only if) the replacement of a subsentence by an L-equivalent one
changed the truth-value and hence also the intension of the whole sen-
tence. In our systems this cannot occur; every sentence in Si (and like-
wise in S 3 , to be explained later) is extensional, and every sentence in Sj
is either extensional or intensional. However, it is the case for a very im-
portant kind of sentence with psychological terms, like ‘ I believe that it
will rain’. Although sentences of this kind seem to be quite clear and un-
problematic at first glance and are, indeed, used and understood in every-
day life without any difficulty, they have proved very puzzling to logicians
who have tried to analyze them. Let us see whether we can throw some
light upon them with the help of our semantical concepts.

In order to formulate examples, we take here, as our object language S,
not a symbolic system but a part of the English language. We assume that
5 is similar in structure to Si except for containing the predicator ‘ . . be-
lieves that - and some mathematical terms. We do not specify here the
rules of S; we assume that the semantical rules of 5 are such that the
predicator mentioned has its ordinary meaning; and, further, that our
semantical concepts, especially ‘true’, ‘L-true’, ‘equivalent’, and ‘L-
equivalent’, are defined for S in accord with our earlier conventions. Now
we consider the following two belief-sentences; ‘D’ and'D" are here writ-
ten as abbreviations for two sentences in S to be explained presently :

(i) ‘John believes that D’.

(ii) ‘John believes that D”.

Suppose we examine John with the help of a comprehensive list of sen-
tences which are L-true in S ; among them, for instance, are translations
into English of theorems in the system of [P.M.] and of even more com-
plicated mathematical theorems which can be proved in that system and
therefore are L-true on the basis of the accepted interpretation. We ask

54 I. THE METHOD OF EXTENSION AND INTENSION

John, for every sentence or for its negation, whether he believes what it
says or not. Since we know him to be truthful, we take his affirmative or
negative answer as evidence for his belief or nonbelief. Among the simple
L-true sentences, there will certainly be some for which John professes
belief. We take as ‘ D’ any one of them, say ‘ Scott is either human or not
human’. Thus the sentence (i) is true. On the other hand, since John is a
creature with limited abilities, we shall find some L-true sentences in 5
for which John cannot profess belief. This does not necessarily mean that
he commits the error of believing their negations; it may be that he can-
not give an answer either way. We take as D' some sentence of this kind;
that is to say, D' is L-true but (ii) is false. Thus the two belief-sentences
(i) and (ii) have different truth-values; they are neither equivalent nor
L-equivalent. Therefore, the definitions of interchangeability and L-inter-
changeability (ii-ia) lead to the following two results:

13 - 1 . The occurrence of ‘ D’ within (i) is not interchangeable with ‘ D".
13 - 2 . The occurrence of ‘ D’ within (i) is not L-interchangeable with ‘ D”.

‘D’ and ‘D" are both L-true; therefore: *

13 - 3 . ‘D’ and ‘D” are equivalent and L-equivalent.

Examining the first belief-sentence (i) with respect to its subsentence
‘D’, we see from 13-1 and 13-3 that the condition of extensionality (ii-2a)
is not fulfilled; and we see from 13-2 and 13-3 that the condition of inten-
sionality (ii-3a) is not fulfilled either:

'^ 13 - 4 . The belief-sentence (i) is neither extensional nor intensional with re-
. spect to its subsentence ‘D’.

v^lthough ‘D’ and ‘D” have the same intension, namely, the L-true or
necessary proposition, and hence the same extension, namely, the truth-
value truth, their interchange transforms the first belief-sentence (i) into
the second (ii), which does not have the same extension, let alone the same
intension, as the first.

\^The same result as 13-4 holds also if any other sentence is taken instead
of ‘D’, in particular, any factual sentence.

Let us now try to answer the much-discussed question as to how a
sentence reporting a belief is to be analyzed and, in particular, whether
such a sentence is about a proposition or a sentence or something else. It
seems to me that we may say, in a certain sense, that (i) is about the
sentence ‘D’, but also, in a certain other sense, that (i) is about the propo-
sition that D. In interpreting (i) with respect to the sentence'D’, it would,
of course, not do to transform it into ‘John is disposed to an affirmative

§ 13. SENTENCES ABOUT BELIEFS SS

response to the sentence ‘D”, because this might be false, although (i)
was assumed to be true; it might, for instance, be that John does not un-
derstand English but expresses his belief in another language. Therefore,
we may try the following more cautious formulation :

(iii) ‘John is disposed to an affirmative response to some sentence in
some language, which is L-equi valent to ‘D’ ’.

Analogously, in interpreting (i) with respect to the proposition that D, the
formulation ‘John is disposed to an affirmative response to any sentence
expressing the proposition that D’ would be wrong because it implies that
John understands all languages. Even if the statement is restricted to
sentences of the language or languages which John understands, it would
still be wrong, because ‘D'’, for example, or any translation of it, likewise
expresses the proposition that D, but John does not give an affirmative
response to it. Thus we see that here again we have to use a more cautious
formulation similar to (iii) :

(iv) ‘John is disposed to an affirmative response to some sentence in
some language which expresses the proposition that D’.

However, it seems to me that even the formulations (iii) and (iv), which
are L-equivalent, should not be regarded as anything more than a first
approximation to a correct interpretation of the belief-sentence (i). It is
true that each of them follows from (i), at least if we take ‘ belief’ here in
the sense of ‘expressible belief’, leaving aside the problem of belief in a
wider sense, interesting thougli it may be. However, (i) does not follow
from either of them. This is easily seen if we replace ‘ D’ by ‘ D”. Then (iii)
remains true because of 13-3; on the other hand, (i) becomes (ii), which is
false. It is clear that we must interpret (i) as saying as much as (iii) but
something more; and this additional content seems difficult to formulate.
If (i) is correctly interpreted in accord with its customary meaning, then
it follows from (i) that there is a sentence to which John would respond
affirmatively and which is not only L-equivalent to ‘D’, as (iii) says, but
has a still stronger relation to ‘D’ — in other words, a sentence which has
something more in common with ‘D’ than the intension. The two sen-
mnre.s mus t, so to speak, be o n^pr<itnnd:.m the sarp e^w^ ; they must not
only be L-equ ivalent in the whole but consist of L-equ ivalent parts, and
both must be built up out of these narfs in thp tmmp If thi^ is the
case, we shall sav that the two sentences have the same intensional struc-
ture^ T his concept will be explicated in the next section and applied in the
analysis of belief sentences in § 15.

S6 I. THE METHOD OP EXTENSION AND INTENSION

§ 14. Intensional Structure

If two sentences are built in the same way out of designators (or designator
matrices) such that any two corresponding designators are L-equivalent, then
we say that the two sentences are inlensionally isomorphic or that they have the
same inknsional structure. The concept of L-equi valence can also be used in a
wider sense for designators in different language systems; and the concept of
intensional isomorphism can then be similarly extended.

We shall discuss here what we call the analysis of the intensional struc-
tures of designators, especially sentences. This is meant as a semantical
analysis, made on the basis of the semantical rules and aimed at showing,
say for a given sentence, in which way it is built up out of designators
and what are the intensions of these designators. If two sentences are built
in the same way out of corresponding designators with the same inten-
sions, then we shall say that they have the same intensional structure.
We might perhaps also use for this relation the term ‘synonymous’, be-
cause it is used in a similar sense by other authors (e.g., Langford, Quine,
and Lewis), as we shall see in the next section. We shall now try to expli-
cate this concept.

Let us consider, as an example, the expressions ‘ 2 -f- 5’ and ‘II sum V’
in a language S containing numerical expressions and arithmetical func-
tors. Let us suppose that we see from the semantical rules of 5 that both
‘ and ‘sum’ are functors for the function Sum and hence are L-equiva-
lent; and, further, that the numerical signs occurring have their ordinary
meanings and hence ‘ 2’ and ‘II’ are L-equivalent to one another, and like-
wise ‘5’ and' V’.VIien we shall say that the two expressions are intension-
ally isomorphic or that they have the same intensional structure, because
they not only are L-equivalent as a whole, both being L-equivalent to
‘7’, but consist of three parts in such a way that corresponding parts are
L-equivalent to one another and hence have the same intensioilfNow it
seems advisable to apply the concept of intensional isomorphism in a
somewhat wider sense so that it also holds between expressions like
‘2-1-5’ and ‘sum(II,V)’, because the use in the second expression of a
functor preceding the two argument signs instead of one standing between
them or of parentheses and a comma may be regarded as aI^^essential
syntactical device. Analogously, if ‘ >’ and* Gr’ are L-equivalent, and like-
wise ‘3’ and ‘III’, then we regard ‘5 > 3’ as intensionally isomorphic to
‘Gr(V,III)’. Here again we r^ard the two predicators ‘ >’ and ‘Gr’ as
/torresponding to each other'I'Irrespective of their places in the sentences;
further, we correlate the first argument expression of ‘ > ’ with the first of
‘Gr’, and the second with the second. Further, ‘2 -f 5 > 3’ is isomorphic

§ 14. INTENSIONAL STRUCTURE

57

to ‘Gr[sum(II,V),IIIl’, because the corresponding expressions ‘2 + S’
and ‘sum (II, V)’ are not only L-equivalent but isomorphic. On the other
hand, ‘7 > 3’ and ‘Gr[sum(II,V),III]’ are not isomorphic; it is true that
here again the two predicators ‘ > ’ and ‘ Gr’ are L-equivalent and that cor-
responding argument expressions of them are likewise L-equivalent^'^t
the corresponding expressions ‘7’ and ‘sum(II,V) are not isomorphic.^^e
require for isomorphism of two expressions that the analysis of both down
to the smallest subdesignators lead to analogous results.

We have said earlier (§ i) that it seems convenient to take as designa-
tors in a system 5 at least all those expressions in 5 , but not necessarily
only those, for which there are corresponding variables in the metalan-
guage M. For the present purpose, the comparison of intensional structures,
it seems advisable to go as far as possible and take as designators all those
expressions which serve as sentences, predicators, functors, or individual
expressions of any type, irrespective of the question of whether or not M
contains corresponding variables. Thus, for example, we certainly want
to regard as isomorphic ‘ q' and ‘ where ‘A’ is the sign of disjunc-
tion (or alternation) as used by the Polish logicians in their parenthesis-
free notation, even if M, as is usual, does not contain variables of the type
of connect! vesVWe shall then regard ‘V’ and ‘A’ as L-equivalent connec-
tiveV^cause any two full sentences of them with the same argument ex-
pressions are L-equivalent.

Frequently, we want to compare the intensional structures of two ex-
pressions which belong to different language systems. This is easily pos-
sible if the concept of L-equivalence is defined for the expressions of both
languages in such a way that the following requirement is fulfilled, in
analogy to our earlier conventions; an expression in 5 is L-equivalent to an
expression in S' if and only if the semantical rules of S and S' together,
without the use of any knowledge about (extra-linguistic) facts, suffice
to show that the two expressions have the safne extension. Thus, L-
equivalence holds, for example, between ‘a’ in 5 and ‘a'’ in S' if we see
from the rules of designation for these two individual constants that both
stand for the same individual ; likewise between ‘ P’ and ‘ P", if we see from
the rules alone that these predicators apply to the same individuals; be-
tween two functors ‘ -t- ’ and ‘ sum’, if we see from the rules alone that they
assign to the same arguments the same values — in other words, if their
full expressions with L-equivalent argument expressions (e.g., ‘2 -|- 5’
and ‘sum(II,V)’) are L-equivalent; for two sentences, if we see from the
rules alone that they have the same truth-value (e.g., ‘Rom ist gross’ in

58

I. THE METHOD OF EXTENSION AND INTENSION

German, and ‘Rome is large’ in English). Thus, even if the sentences
‘2 + S > 3’ and'Gr [sum(II,V),III]' belong to two different systems, we
find that they are intensionally isomorphic by establishing the L-equiva-
lence of corresponding signs.

Vlf variables occur, the analysis becomes somewhat more complicated,
but the concept of isomorphism can still be defined. We shall not give here
exact definitions but merely indicate, with the help of some simple ex-
amples, the method to be applied in the definitions of L-equivalence and
isomorphism of matrices. Let ‘a:’ be a variable in S which can occur in a
universal quantifier ‘(x)’ and also in an abstraction operator ‘(Xa:)’, and
‘m’ be a variable in S' which can occur in a universal quantifier ‘IIm’ and
also in an abstraction operator ‘iJ’. If ‘x’ and ‘u’ have the same range of
values (or, more exactly, of value intensions, § 10), for example, if both
are natural number variables (have natural number concepts as value in-
tensions), we shall say that ‘x’ and ‘ u’ are L-equi valent, and also that
'(*)’ and ‘n«’ are L-equivalent, and that‘(Xx)’ and'Jj’ are L-equivalent.
If two matrices (sentential or other) of degree n are given, one in S and
the other in S', we say that they are L-equivalent with respect to a certain
correlation between the variables, if corresponding abstraction expressions
are L-equivalent predicators. Thus, for example, ‘ X > y’in 5 and‘Gr(M,D)’
in S' are L-equivalent matrices (with respect to the correlation of
with ‘u’ and ‘y' with ‘v’) because '{'kxy) [x > y]’ and ' ^v[Gt{u,v)]’ are
L-equivalent predicators. Intensional isomorphism of (sentential or other)
matrices can then be defined in analogy to that of closed designators, so
that it holds if the two matrices are built up in the same way out of cor-
responding expressions which are either L-equivalent designators or L-
equivalent matrices. Thus, for example, the matrices ‘x ^ > y’ and
‘ Gr[sum(M,V),i>]’ are not only L-equivalent but also intensionally isomor-
phic; and so are the (L-false) sentences ‘(a;) (>’)[a: + 5 > y\' and ‘HuTlv
[Gr[sum(«,V),J']]’.

These considerations suggest the following definition, which is recur-
sive with respect to the construction of compound designator matrices
out of simpler ones. It is formulated in general terms with respect to
designator matrices; these include closed designators and variables as
special cases. The definition presupposes an extended use of the term ‘L-
equivalent’ with respect to variables, matrices, and operators, which has
been indicated in the previous examples but not formally defined. The
present definition makes no claim to exactness; an exact definition would
have to refer to one or two semantical systems whose rules are stated
completely.

§ 15. APPLICATIONS OF INTENSIONAL STRUCTURE

59

14-1^ Definition of intensional isomorphism

a. Let two designator matrices be given, either in the same or in two
different semantical systems, such that neither of them contains an-
other designator matrix as proper part. They are intensionally iso-
morphic =Df they are L-equivalent.

b. Let two compound designator matrices be given, each of them con-
sisting of one main submatrix (of the type of a predicator, functor,
or connective) and n argument expressions (and possibly auxiliary
signs like parentheses, commas, etc.). The two matrices are inten-
sionally isomorphic =d{ (i) the two main submatrices are intension-
ally isomorphic, and [ 2 ) for any m from i to n, the wth argument
expression within the first matrix is intensionally isomorphic to the
mth in the second matrix (‘ the wth’ refers to the order in which
the argument expressions occur in the matrix).

c. Let two compound designator matrices be given, each of them con-
sisting of an operator (universal or existential quantifier, abstrac-
tion operator, or description operator) and its scope, which is a
designator matrix. The two matrices are intensionally isomor-
phic = Df (i) the two scopes are intensionally isomorphic with re-
spect to a certain correlation of the variables occurring in them, ( 2 )
the two operators are L-equivalent and contain correlated variables.

In accord with our previous discussion of the explicandum, rule b in
this definition takes into consideration the order in which argument ex-
pressions occur but disregards the place of the main subdesignator. For
the intensional structure, in contrast to the merely syntactical structure,
only the order of applicatio ajs essenti al, not the order and manner of

§ 16. Applications of the Concept of Intensional Structure

The concept of intensional structure is compared with the concepts of
synonymity discussed by Quine and Lewis. The concept is then used for giving
an interpretation of belief sentences that seems more adequate than the inter-
pretations discussed earlier (§ 13). Further, the same concept helps in solving
the so-called paradox of analysis.

It has often been noticed by logicians that for the explication of certain
customary concepts a stronger meaning relation than identity of intension
seems to be required. But usually this stronger relation is not defined. It
seems that in many of these cases the relation of intensional isomorphism
could be used. For example, if we ask for an exact translation of a given
statement, say the exact translation of a scientific hypothesis or of the

60

I. THE METHOD OF EXTENSION AND INTENSION

testimony of a witness in court from French into English, we should usual-
ly require mnrp thaj; agreement in ^e inten sions of the sentences,
that is, L-eq ui valence of the sentenc es. Even if we restrict our attention to
designative (cognitive) meaning — leaving aside other meaning compo-
nents like the emotive and the motivative, although they are often very
important even for the translation of theoretical texts — L-equivalence of
sentences is not sufficient; it will be required that at least some of the com-
ponent designators be L-equivalent, in other words, that the intensional
structures be alike or at least similar.

Quine explains, without giving a definition, a concept of synonymity
which is different from and presumably stronger than L-equivalence. He
says: '*^he notion of synonymity figures implicitly also whenever we use
the method of indirect quotations. In indirect quotation we do not insist
on a literal repetition of the words of the person quoted, but we in^i^ on
a synonymous sentence ^e require reproduction of the meaning. Such
synonymity differs even from logical equivalence ; and exactly what it is
remains unspecified.”’® We might perhaps think of an explicatum of this
concept of synonymity similar to our concept of intensional isomorphism.
Quine himself seems to expect that the explication will be found not in
semantics but in what we would call pragmatics, because he says that the
concept of synonymity “calls for a definition or a criterion in psycho-
logical and linguistic terms.”

C. I. Lewis” gives a definition for the concept of synonymity which
shows a striking similarity to our concept of intensional isomorphism, al-
though the two concepts have been developed independently. Since it is
interesting to see the points of agreement and of difference, I will quote his
explanations at length. “Not every pair of expressions having the same
intension would be called synonymous; and there is good reason for this
fact. Two expressions are commonly said to be synonymous (or in the case
of propositions, equipollent) if they have the same intension, and that in-
tension is neither zero nor universal^^ut to say that two expressions with
the same intension have the same meaning, without qualification, would
have the anomalous consequence thafahy two cuialytic propositions would
then be equipollent, and fey two self-contradictory propositions would be
equipollent.” In order to overcome this difficulty, Lewis introduces a new
concept: “ Two expressions are equivalent in analytic meaning, (i) if at least
one is elementary [i.e., not complex] and they have the same intension, or
(2) if, both being complex, they can be so analyzed into constituents that

’• [Notes], p. 120.

19 [Meaning], pp. 245 f. Other concepts used by Lewis will be discussed in the next section.

S IS. applications of intensional structure

6i

V.(a) for every constituent distinguished in either, there is a correspruiding
constituent in the other which has the same intension, (ft) no constituent
distinguished in either has zero intension or universal intension, and (c)
the order of corresponding constituents is the same in both, or can be
made the same without alteration of the intension of either whole expres-
sion.” As examples, Lewis states that “round excision” and “circular
hole” are equivalent in analytic meaning, while “equilateral triangle” and
“equiangular triangle” are not, although they have the same intension.
He continues; “We shall be in conformity with good usage if we say that
two expressions are synonymous or equipollent, (i) if they have the same
intension and that intension is neither zero nor universal, or (2) if, their
intension being either zero or universal, they are equivalent in analytic
meaning.”

Thus Lewis’ concept of synonymity is very similar to our concept of
intensional isomorphism except for one point ?^e applies this stronger
relation only to the two extreme cases of intension, for example, in the
field of sentences, only to L-determinate and not to factual sentences.
This discrimination seems to me somewhat arbitrary and inadvisable. Let
us consider the following examples (in a language which, in distinction to
S„ also contains expressions for finite cardinal numbers and for relations
and properties of them):

(i) ‘two is an even prime number’;

(ii) ‘two is between one and three’;

(iii) ‘ the number of books on this table is an even prime number’ ;

(iv) ‘the number of books on this table is between one and three’.

The sentences (i) and (ii) have the same intension but are not equivalent
in analytic meaning (intensionally isomorphic). The same holds for (iii)
and (iv). Now, according to Lewis’ definition, (i) and (ii) are not synony-
mous because they are L-true, analytic; while'(iii) and (iv) are synony-
mous because they are factual, synthetic. It seems to me that it would be
more natural to regard (iii) and (iv) also as nonsynonymous, .since the dif-
ference between them is essentially the same as that between (i) and (ii).
The logical operation which leads from (i) to (ii) is the same as that which
leads from (iii) to (iv) ; it is the transformation of ‘w is an even prime num-
ber’ into ‘m is (a cardinal number) between one and three’.

Now let us go back to the problem of the analysis of belief-sentences,
and let us see how the concept of intensional structure can be utilized
there. It seems that the sentence ‘John believes that D’ in S can be in-
terpreted by the following semantical sentence ;

6a I. THE METHOD OF EXTENSION AND INTENSION

" 16 - 1 . ‘There is a sentence ©< in a ^mantical system S' such that (a) ©i
is intensionally isomorphic to ‘D’ and (b) John is disposed to an affirma-
tive response to

This interpretation may not yet be final, but it represents a better ap-
proximation than the interpretations discussed earlier (in § 13). As an
example, suppose that John understands only German and that he re-
sponds affirmatively to the German sentence ‘ Die Anzahl der Einwohner
von Chicago ist grosser als 3,000,000’ but neither to the sentence ‘Die
Anxahl der Einwohner von Chicago ist grosser als 2^ X 3 X 5®’ nor to
any intensionally isomorphic sentence, because he is not quick enough to
realize that the second sentence is L-equivalent to the first. Then our
interpretation of belief-sentences, as formulated in 15-1, allows us to assert
the sentence ‘John believes that the number of inhabitants of Chicago is
greater than three million’ and to deny the sentence ‘John believes that
the number of inhabitants of Chicago is greater than 2* X 3 X j'’’. We
can do so without contradiction because the two German sentences, and
likewise their EngUsh translations just used, have different intensional
structures. [By the way, this example shows another disadvantage of
Lewis’ definition of equivalence in analytic meaning. According to part
(i) of his definition, the two German sentences are equivalent in analytic
meaning if we take '3,ooo,c»oo’ as one sign.] On the other hand, the in- ,
terpretation of belief-sentences in terms of propositions as objects of be-
liefs (like (iv) in § 13) would not be adequate in this case, since the two
German sentences and the two English sentences all express the same
proposition.

An analogous interpretation holds for other sentences containing psy-
chological terms about knowledge, doubt, hope, fear, astonishment, etc.,
with ‘ that’-clauses, hence generally about whaflfussell calls propositional
attitudes and Ducasse epistemic attitudes.'’^^he problem of the logical
analysis of sentences of this kind has been much discussed, but a satis-
factory solution has not been found so farVthe analysis here proposed is
not yet a complete solution, but it may perhaps be regarded as a first step.
What remains to be done is, first, a refinement of the analysis in terms of
linguistic reactions here given and, further, an analysis in terms of dispo-
sitions to nonlinguistic behavior.

Russell, [Inquiry], gives a detailed discussion of the problem in a wider sense, including
beliefs not expressed in language; he investigates the problem under both an epistemological
and a logical aspect (in our terminology, both a pragmatical and a semantical aspect), not
always distinguishing the two clearly. For C. J. Ducasse’s conception see his paper “Proposi-
tions, Opinions, Sentences, and Facts,” Journal of Philosophy, XXXVII (1940), 701-11.

6 IS. APPLICATIONS OF INTENSIONAL STRUCTURE 63

The concept of intensional structure may also help in clarifying a
puzzling situation that has been called “the paradox of analysis”. It was
recently stated by G. E. Moore/' and then discussed by C. H, Langford,^
Max Black/’ and Morton White. ■*♦ Langford^® states the paradox as fol-
lows: “If the verbal expression representing the analysandum has the
same meaning as the verbal expression representing the analysans, the
analysis states a bare identity and is trivial; but if the two verbal expres-
sions do not have the same meaning, the analysis is incorrect.” Consider
the following two sentences:

‘The concept Brother is identical with the concept Male Sibling.’

‘The concept Brother is identical with the concept Brother.’

The first is a sentence conveying fruitful information, although of a
logical, not a factual, nature; it states the result of an analysis of the
analysandum, the concept Brother. The second sentence, on the other
hand, is quite trivial. Now Moore had been puzzled by the following fact:
If the first sentence is true, then the second seems to make the same
statement as the first (presumably because, if two concepts are identical,
then a reference to the one means the same as a reference to the other,
and hence the one expression can be replaced by the other); “but it is
obvious that these two statements are not the same”, he says. Black tries
to show that the two sentences do not express the same proposition; he
supports this assertion by pointing to the fact that the first sentence, or
rather a paraphrasing he gives for it (‘the concept Brother is the conjunct
of the concept Male and the concept Sibling’) refers to a certain non-
identical relation (the triadic relation Conjunct), while the second is a
mere identity. White replies that this is not a sufficient reason for the as-
sertion. None of the four authors states his criterion for the identity of
“meaning”, “statement”, or “ proposition” ; this seems the chief cause for
the inconclusiveness of the whole discussion. If we take, as in the terminol-
ogy used in this book, L-equivalence as the condition for the identity of
propositions, then White is certainly right; since the two sentences are
L-true and hence L-equivalent to each other, they express the same
proposition in our sense. On the other hand. Black feels correctly, like
Moore and Langford, that there is an important difference in meaning
idSetween the two sentences, because of a difference in meaning between

The Philosophy of G. E. Moore, ed. P. Schiipp (1942), pp. 660-67.

« “The Notion of Analysis in Moore’s Philosophy”, ibid., pp. 321-42.

Mind, LIII (1944), 263-67 and LIV (1945), 272 f.

** Mind, LIV (1945), 71 f. and 357-61. « Op. cit., p. 323.

64 I. THE METHOD OF EXTENSION AND INIENSION

the two expressions for the analysandum (‘ the concept Brother') and the
analysans (‘the concept Male Sibling’) .'^e paradox can be solved if we
can state exactly what this difiFerence in meaning is and how com-
patible with the identity of meaning in another senseVfllie solution is
quite simple in terms of our conceptsW'he difference between the two
expressions, and, consequently, between the two sentences is a difference
in intensional structure, which exists in spite of the identity of intension.
Langford saw the point at which the difference lies; he says^* that the
analysans is more articulate than the analysandum, it is a grammatical
function of more than one idea; the two expressions are not synonymous
but “cognitively equivalent in some appropriate sense”. It seems to me
that this cognitive equivalence is explicated by our concept of L-equiva-
lence and that the synonymity, which does not hold for these expressions,
is explicated by intensional isomorphism.

§ 16. Lewis’ Method of Meaning Analysis

Lewis uses, in addition to the concepts of extension and intension which are
similar to ours, the concept of comprehension which presupposes the admission
of nonactual, possible things. It seems inadvisable to use this conception be-
cause it requires a new, more complicated language form. The distinction which
Lewis wants to make can better be made with respect to intensions than with re-
spect to things.

I wish to discuss briefly some concepts which have recently been pro-
posed by C. I. Lewis^^ as tools for a semantical meaning analysis. There
is a striking similarity between these concepts and our concepts of exten-
sion and intension. This similarity is due to the common aim to mtike some
traditional concepts, especially extension and intension, denotation and
connotation, more general in their application and, at the same time,
more clear and precise.

Lewis explains his chief semantical concepts in the following way: “All
terms have meaning in the sense or mode of denotation or extension;
and all have meaning in the mode of connotation or intension. The denota-
tion of a term is the class of all actual or existent things to which that
term correctly applies. . . . The comprehension of a term is the classifica-
ct/., p. 326.

In [Meaning]. This paper is part of a “S3Tnposiura on Meaning and Truth”, published
in four parts in Philosophy and Phenomenological Research, Vols. IV (1943-44) and V (1944-
4S). This symposium also contains a number of other interesting contributions to the de-
velopment and clarification of semantical concepts. I have elsewhere referred to Tarski’s
paper [Truth]; I am in close agreement with his conception of the nature of semantics, but
he does not discuss the central problems of this book. Concerning these problems, I wish
especially to call attention to the papers by C. J. Ducasse (IV, 317-40; V, 320-32) and Charles
A. Baylis (V, 80-93).

{ 16. LEWIS’ METHOD OF MEANING ANALYSIS 65

tion of all consistently thinkable things to which the term would correctly
apply. . . . For example, the comprehension of “square” includes all
imaginable as well as all actual squares, but does not include round
squares. . . . The connotation or intension of a term is delimited by any
correct definition of it.”

It seems that Lewis’ concepts of extension and intension correspond
closely to our concepts. This is clearly the case for predicators, but per-
haps also for sentences and individual expressions. There remains the
problem of the necessity and usefulness of Lewis’ third concept, that of
comprehension. It seems that Lewis follows Meinong^* in dividing (i) all
things (in the widest sense) into impossible or inconceivable things (e.g.,
round squares) and possible things; and (2) the possible things into actual
things (e.g., Plato) and nonactual possible things (e.g., Apollo, unicorns).
[Lewis clearly makes the second division. Whether he also makes the first
and hence countenances, like Meinong, impossible things is not quite so
clear but seems indicated by the formulation that the comprehension
“does not include round squares”. According to the ordinary conception,
in distinction to Meinong’s, there are no round squares at all, not even in
some particular kind of objects; hence it would be redundant to say that
the comprehension “does not include round squares”.] Meinong’s concep-
tion has been critically discussed by RusselP’ and then rejected. Russell’s
chief reason for the rejection is that the impossible objects violate the
principle of contradiction; for example, a round square is both round and
nonround, because square. Russell is certainly right in the following re-
spect: Within the logical framework of our ordinary language, we cannot
consistently apply the conception of impossible things or even that of
possible nonactual things. And, as far as I am aware, neither Meinong nor
Lewis nor any other philosopher has constructed or even outlined a
language of a new structure which would accommodate those entities.
That such a language must be different from the ordinary one is shown by
the following example; In the ordinary language we say: ‘There are no
white ravens and no round squares’. In the new language we would have to^
say, instead; '^here are white ravens; however, they are not actual, but
only possibleV'And there are round squares; however, they are neither'"
actual nor possible, but impossible.’ I have no doubt that a resourcefut
logician could easily construct a consistent language system of this kind,
if we wanted it; he would have to lay down rules for the quantifiers deviat-
ing from the ordinary rules in a way suggested by the examples. The

<• A. von Meinong, Untersuchungen zur Gegenstandstheorie und Psychologie (1904).

« [Denoting], pp. 482 f.

66 I. THE METHOD OF EXTENSION AND INTENSION

decisive question is not that of the technical possibility of such a language
but rather that of its usefulness. Only if it can be shown to have great ad-
vantages in comparison to the ordinaiy language structure would it be
worth considering in spite of its fundamental deviation and increased
complexity.

*'! do not see sufficient reasons for this change. The distinctions which
Meinong and Lewis have in mind are important, but they can be taken
care of in a different way. Instead of dividing objects into (i) actual, (ii)
nonactual but possible, and (iii) impossible, we make analogous distinc-
tions, first, between three corresponding kinds of expressions and then be-
tween three corresponding kinds of intensions. Let us show this, first, for
predicators. Instead of speaking about three kinds of objects like this:

(i) ‘ (some) horses are actual objects’,

(ii) ‘unicorns are nonactual but possible objects’,

(iii) ‘round squares are impossible objects’,

jre speak, rather, about three kinds of predicators :

(i) ‘ the predicator ‘ horse’ is not empty’,

(ii) ‘ the predicator ‘ unicorn’ is F-empty, i.e., empty but not L-empty ’,

(iii) ‘the predicator ‘ round square’ is L-empty’.

Then we apply the same terms to the corresponding intensions (this is a
transference of terms from a semantical to a nonsemantical use, analogous
to the transference of the terms ‘equivalent’ and ‘L-equivalent’, § 5):

(i) ‘the property Horse is not empty’,

(ii) ‘the property Unicom is F-empty, i.e., empty but not L-empty’,

(iii) ‘the property Round Square is L-empty’.

An analogous distinction can be made for individual expressions, for
instance, descriptions. (We apply here, not the special interpretation of
descriptions which we adopted in § 8 because of its technical advantages,
but the customary interpretation, according to which a description has a
descriptum only if the uniqueness condition is fulfilled.) Then, instead of
using the following formulations referring to objects:

(i) ‘Alexander’s horse (i.e., the one horse which Alexander had at such
and such a time) is an actual object’,

(ii) ‘Alexander’s unicorn is a nonactual but possible object’,

(iii) ‘Alexander's round square is an impossible object’,

we use, rather, the following ones concerning individual expressions

(Lewis’ singular terms) :

§ 16. LEWIS’ METHOD OF MEANING ANALYSIS 67

(i) ‘the description ‘Alexander’s horse’ is not empty’,

(ii) ‘the description ‘Alexander’s unicorn’ is F-empty’ (in Lewis’
terminology, it has zero denotation, but not zero comprehension) ;

(iii) ‘the description ‘Alexander’s round square’ is L-empty’ (it has
zero comprehension).

And then we make analogous statements concerning the corresponding
individual concepts (in Lewis’ terminology, connotations of singular
terms) :

(i) ‘the individual concept Alexander’s Horse is not empty’,

(ii) ‘ the individual concept Alexander’s Unicom is F-empty’,

(iii) ‘the individual concept Alexander’s Round Square is L-empty’.

Thus our method does not neglect the distinctions pointed out by Mei-
nong and Lewis. However, it applies the distinction to intensions, while
these philosophers apply it to objects and thereby violate the rule of ordi-
tnary language Vtuch takes the addition of ‘ actual’ to a general noun as
redundant. For example, the ordinary language takes phrases like ‘actual
horses’, ‘real horses’, ‘existing horses’, etc. (where ‘actual’, etc., does not
mean ‘occurring at t^ present time’ but ‘occurring at some time, past,
present, or future’), as meaning the same as ‘horses’, differing from this
only in emphasis; and)Tikewise, ‘ actual unicorns’ is taken as meaning the
same as ‘unicorns’, and hence it is said: ‘there are no unicorns (at any
space- time point)’.

If we thus reject such distinctions between kinds of objects, then Lewis’
concept of comprehension can no longer be defined. Do we hereby sacrifice
a useful tool of semantical meaning analysis? I do not think so. Lewis
emphasizes rightly the difference between comprehension and extension.
But there seems not to be much difference between the purposes of the
concepts of comprehension and intension. If we accept Lewis’ language
form, then these concepts are both legitimate and, of course, not identical.
But whatever is said in terms of comprehension can immediately be trans-
lated into terms of intension, because comprehension and intension deter-
mine each other logically. If you tell me the comprehension of a Chinese
word, then I know immediately what is its intension, and vice versa;
therefore there is no advantage in having both concepts. On the other
hand, if you tell me the intension of a Chinese word, I do not know its
extension (unless it is L-determinate) ; and if you tell me only its exten-
sion, I cannot infer from this its intension. Therefore, it is useful to have
both concepts, that of intension and that of extension.

We also arrive at the same result, the rejection of nonactual, possible

I. THE METHOD OF EXTENSION AND INTENSION

objects and of comprehension by an. approach from another angle, that of
modal logic. We shall find later (§§ 42 f.) that the logical modalities must
be applied to intensions, not to extensions. Thus we may speak of an im-
possible (or L-false) proposition but not of an impossible truth-value; of an
impossible (or L-empty) property but not of an impossible (or L-empty)
class. Analogously, we may speak of an impossible (or L-emptyVifidi-
vidual concept but not of an impossible individual (object, thing)VlSecause
jodiuduaJa. ^iHngs} -are extensions,- not- intp-nainna; in other

words, individu als a re involved in guestions of application , not in ques-
t ions of meaning in the strict sense. (We take here, of course, the ordinary
conception of ^tensions, not that to be discussed in § 23, according to
which extensions are construed as a special kind of intension.)

To sum up, I do not think that the concepts of possible and impossible
objects and of comprehension can be accused of violating logic or of lead-
ing necessarily to contradictions. However, it seems doubtful whether
these concepts are sufficiently useful to compensate for their disadvantage
— the necessity of using an uncustomaiy and more complex language
structure.

CHAPTER II

L-DETERMINACY

We have seen (§ 2) that a sentence is L-determinate if its truth-value, which
is its extension, is determined by the semantical rules. In this chapter we apply
the concept of L-determinacy also to other designators. The definitions are con-
structed so that an analogous result holds; A designator is L-determinate if the
semantical rules, independently of facts, suffice for determining its extension
(§17). For the application of this concept we presuppose that the individuals
are positions in an ordered domain. An individual expression is L-determinate
if the semantical rules suffice for determining the location of the position to which
it refers (§§ 18, ig). A predicator is L-determinate if the semantical rules suf-
fice for determining for every position whether the predicator applies to it or
not (§20). The distinction between logical and descriptive (nonlogical) signs
is discussed, and its connection with the distinction between L-determinate and
L-indeterminate designators is examined {§ 21). The intension of an L-determi-
nate designator is also called L-determinate (§ 22). There is a one-one correla-
tion between extensions and L-determinate intensions; therefore, it would be
possible, though not customary, to define extensions as L-determinate inten-
sions (§23).

§ 17 . L-Determinate Designators

In general, factual knowledge is needed for establishing the truth-value of a
given sentence. However, if the sentence is L-determinate (§ 2), the semantical
rules suffice for establishing its truth-value or, in other words, its extension.
The concept of L-determinacy will now be extended to designators of other
kinds. We stipulate that the definitions of this concept for the other kinds be
such that a designator is L-determinate if and only if the semantical rules suffice
for determining its extension. Definitions fulfilling this requirement will be con-
structed in later sections of this chapter.

We found earlier that the intension of the sentence ‘Hs’ in the system Si
is the proposition that Scott is human and that its extension is the truth-
value truth. Now let us consider the question of what knowledge we need
in this and other cases in order to determine the intension and the exten-
sion of a given sentence. It is clear that, for the determination of the in-
tension, only the semantical rules of the system Si are required. For every
sentence in Si these rules give an interpretation and thereby tell us what
proposition is the intension of the sentence. Thus the result mentioned
concerning the intension of ‘ Hs’ is established on the basis of those rules
which give an interpretation for ‘ Hs’ ; these are the rules of designation for
‘H’ and for ‘s’ (see i-i and 1-2) and the rule of truth for atomic sentences
(1-3). On the other hand, for the determination of the extension, the truth-
value, of ‘Hs’, knowledge of the semantical rules alone is obviously not

69

70

II. L-DETERMINACY

STiffident. We need, in addition, factual knowledge. This factual knowl-
edge is based on observations of the thing Walter Scott; these observa-
tions lead to the result that this thing has the properties characteristic of
human beings and, hence, that the sentence ‘ Hs’ is true.

However, we have seen that there is a particular kind of sentence for the
determination of whose truth-values the semantical rules without any
factual knowledge provide a sufficient basis. These are the L-determinate
sentences, that is, the L-true and the L-false sentences (see the explana-
tion preceding 2-7). Thus, for these sentences the semantical rules suf-
fice to determine not only their intensions but also their extensions. Now
we shall extend the meaning of the term ‘L-determinate’ so as to make it
applicable to designators in general, in analogy to its application to sen-
tences. For this purpose it seems natural to lay down the following con-
vention for any semantical system S:

17 - 1 . A designator is L-determinate in 5 if and only if its extension can be
determined on the basis of the semantical rules of 5 alone, without any
reference to facts.

This convention is not itself a definition of ‘L-determinate’. It is meant
merely as an informal characterization of the explicandum ; in other words,
a requirement which the definition should fulfil. A definition of L-deter-
rainacy for sentences has already been given (2-3d). The problems of con-
structing definitions of L-determinacy for other kinds of designators will
be discussed in the subsequent sections. But even if it is regarded as
merely a requirement, the present formulation in 17-1 is found upon
examination to be insufficient. The phrase “ the extension is determined
by certain rules” can be understood in two quite different senses. We have
to find out which sense is appropriate here.

The difficulty here involved can perhaps best be made clear in the case
of a predicator. The intension of the predicator ‘ H’ can obviously be de-
termined with the help of the semantical rules alone; we see from the rule
of designation for ‘H’ (1-2) that its intension is the property Hiunan. But
does the same not hold for the extension, too? Do we not also see from the
same rule that the extension of ‘H’ is the class Human? Should we then
say, according to our convention, that ‘H’, and likewise every other
predicator, is L-determinate? This would obviously not be in accordance
with the intended meaning of this term.

In order to overcome this difficulty, we have to make a certain distinc-
tion which can easily be explained for sentences and then transferred to
designators of other kinds. Suppose we ask the question : “ What is the

§ 17. L-DETERMINATE DESIGNATORS 71

extension, that is, the truth-value, of the sentence ‘Hs'?” Consider the
following sentences under 17-2 and 17-3, which belong to the metalan-
guage M. Let us examine whether they may be regarded as satisfactory
answers to our question.

17 - 2 . a. ‘The extension of ‘Hs’ is the truth- value truth.’

b. ‘ ‘Hs’ is true.’

c. ‘Scott is human.’

d. ‘The extension of ‘Hs’ is the same as that of ‘H = H’.’

e. ‘ ‘Hs’ is equivalent to ‘H = H’.’

17 - 3 . a. ‘The extension of ‘Hs’ is the truth-value that Scott is human.’

b. “Hs’ is true if and only if Scott is human.’

Each of these seven sentences is true (see 6-3). And in some sense each
of them may be said to give an answer to our question. However, there is
an important difference between the sentences under 17-2 and those under
17-3. Suppose we understand the sentences of the system S, but have no
factual knowledge concerning the things referred to in these sentences;
then we do not know whether ‘Hs’ is true or not, in other words, whether
Scott is human or not. Suppose, further, that the purpose of our question
was to find this lacking knowledge. Then i7-2a is a completely satisfac-
tory answer because it supplies the information we want; and so is 2b,
which is merely a simpler formulation for 2a; and likewise 2C, which gives
the same information without the use of semantical terms. (For the result
that 2b and 2c mean the same, see the explanation preceding 1-7.) On the
other hand, the answer 3a, although correct, does not satisfy our purpose;
we shall reply with a modified formulation of our first question; “Yes; but
what is the truth-value that Scott is human?” Similarly, we shall reply to
3b: “Yes; but is Scott human or not?” We may formulate this difference
by saying that 2a, 2b, and 2c actually give the truth- value of ‘ Hs’, while 3a
and 3b do not give it but merely describe it, in the sense of supplying a de-
scription for it (in Russell’s sense of ‘description’). We can do this by
introducing the phrase ‘gives the truth-value’ in the following way, which
is not meant as an exact definition. Let be a true sentence in M (it
may also be a definition or a rule or a set of true sentences, definitions, or
rules). We shall say that the truth- value of a sentence ©< in a system 5 is
given by 0 , if either the sentence ‘ 0 , is true (in S)’ or its negation follows
from 0 , (in M) without the use of any factual knowledge not supplied by

0 ,-. [The phrase ‘. . . follows from (in M)’ may be understood as

meaning the same as ‘ ... is L-implied by (in M)’, if we assume that

L-terms with respect to M have been defined in the metametalanguage

72

n. L-DETERMINACY

MM. For the sake of simplicity, we use the German letters with subscripts
not only in M for expressions in S but also in MM for expressions in M
and expressions in .S.]

Let us now apply this criterion to the sentences under 17-2 and 17-3.
First, 2b fulfils the criterion in a trivial way ; hence it gives the extension
of ‘Hs’. Furthermore, each of the sentences 2a and 2c, and even 2d and
2e, gives, together with the semantical rules of S,, the truth- value of ‘Hs’,
because 2b follows from each of these sentences, together with the rules.
That 2b follows from 2a is obvious. Further, 2c follows from 2b, together
with the result 3b, which is based on the semantical rules for ‘H’, ‘s’, and
atomic sentences (1-2, i-i, 1-3). Sentence 2d is derived from 2a and there-
by from 2b, together with the result that ‘H s H’ is L-true, which, in
turn, is based on the semantical rules. The same holds for 2e, which is
merely another formulation for 2d, according to the definition 5-1. On
the other hand, either 3a or 3b, together with the semantical rules, does
not give the extension of ‘Hs’ but merely describes it, because for the
derivation of 2b we need here the factual knowledge that Scott is human.

Consider, now, in contrast to ‘Hs’, an L-determinate sentence, for ex-
ample, the L-true sentence ‘s = s’ or the L-false sentence ‘~(s = s)’.
Here no factual sentence ©, is required in addition to the semantical rules
to give the truth-values of these two sentences. The following two sen-
tences in M follow from the semantical rules of S, alone: “s = s’ is true
(in S,)’, ‘ ‘~(s = s)’ is not true (in S,)’.

In analogy to these results for sentences, we now replace the earlier
convention 17-1 by the following:

17 - 4 . A designator is L-determinate in S if and only if the semantical
rules of 5 alone, without addition of factual knowledge, give its extension.

This again does not yet constitute a definition of ‘L-determinate’ but
only a requirement which the definition should fulfil. For sentences, the
previous definition of L-determinacy (2-3d) is in accord with this conven-
tion on the basis of our explanation of “. . . gives the extension, i.e., the

truth-value, of the sentence ”. Our task will now be to find adequate

definitions of L-determinacy for the other kinds of designators. For each
of these kinds we shall have to consider the conditions under which their
extensions are actually given, not merely described; as in the case of
sentences, “to give the extension’’ will only be informally explained, not
exactly defined. And then the definition of L-determinacy will be con-
structed in such a way that the requirement 17-4 is fulfilled. If a designa-
tor is not L-determinate, we call it L-indeterminate. This term has

§ 18. L-DETERMINACY OF INDIVIDUAL EXPRESSIONS 73

been defined for sentences (2-7); however, in the case of sentences, we
usually use the synonymous term ‘factual’. According to the convention
17-4, a designator is L-indeterminate if its extension can be given only by
a factual statement (in M).

§ 18. The Problem of L-Determinacy of Individual Expressions

The conditions under which an individual expression may be regarded as
L-determinate are examined. An attempt to base the definition of L-determi-
nacy on a distinction between (genuine) proper names and descriptions is aban-
doned as inadequate. The analysis is then applied to a coordinate language S3.
Its individuals are positions in a discrete, linear order, 'o’, ‘o", ‘o"’, etc., are
the so-called standard individual expressions for these positions in their basic
order. Every one of these expressions indicates by its form the location within
the basic order to which it refers; hence it exhibits its own extension and may
be regarded as L-determinate. The same does not, in general, hold for a descrip-
tion (e.g., ‘the one position which is blue and cold’), except when the description
is L-equivalent to a. standard expression (e.g., ‘the one position which is be-
tween o' and 0'"’).

We begin with individual expressions because, as we shall see later, the
solution of the problem of L-determinacy for predicators presupposes the
solution for individual expressions.

In analogy to the earlier question, “What is the truth-value of ‘Hs’?”
we now consider the question, “Which individual is the extension of
‘(1*) (Axw)’?” and possible answers to it. In analogy to the earlier case,
let us imagine that we do not know whether there is exactly one author of
Waverley and, if so, who he is; and that the purpose of our question is to
find out from somebody who does know. Obviously, the answer ‘the exten-
sion of the description mentioned is the author of Waverley’ would i^ot
satisfy us even though it is true ; it is entirely trivial. [Note that, according
to an earlier convention, the phrase "‘the author of Waverley’ is to be
understood as short for ‘ the one individual who is author of Waverley, or
a’" if there are no or several such individuals’.] The answer ‘ the extension
sought is the author of Ivanhoe’ is true and not trivial; but, nevertheless,
it would not satisfy us because it does not supply the specific information
we are looking for; we might say here again that this answer merely de-
scribes the extension but does not give it. The extension is actually and
directly given by the answer ‘ the extension is Walter Scott’. It is indirectly
given by answers like these; ‘the extension of ‘ (ix) (Axw) ' is the same as
the extension of ‘ s’ ’ or ‘ ‘ (ix) (Axw) = s’ is true’ ; from these we obtain
the direct answer with the help of the semantical rule (i-i), which tells
us that the extension of ‘s’ is Walter Scott.

On the basis of these considerations we might perhaps be inclined to

74

n. L-DETERMINACY

propose the following solution: Let us say that the extension of an indi-
vidual expression is given, and not merely described, by ©y if ©y uses a
proper name in M (e.g., ‘ Walter Scott’) or refers to a proper name in 5
(e.g., ‘s’) in distinction to a description. However, it is easily seen that
this does not yet constitute a satisfactory solution. Suppose that ‘x is a
dagger and Brutus used x for killing Caesar’ can be translated into sym-
bols of S, ; this symbolic expression may be indicated by ‘ . . a: . . ’. Suppose
somebody gives to our question, ‘What is the extension of the description
‘ (ix) the answer ‘The extension is the same as the extension of

‘ b’ where ‘ b’ is an individual constant in S, such that ‘ (ix) (. . x . .) s b’
is true. Then the answer is true. According to our tentative solution, we
should say that this answer gives the extension of ‘ (ix) (. . x . .) ’, irrespec-
tive of the way in which the semantical rule for ‘ b’ is formulated. But sup-
pose, now, that this rule says that ‘b’ is the symbolic translation of ‘the
dagger with which Brutus killed Caesar’. Then the above answer says, in
other words, that the extension of ‘ (ix) (. . x . .) ’ is the dagger with which
Brutus killed Caesar; thus the answer merely describes the extension. The
reason for this lies in the fact that the interpretation of the constant ‘ b’ is
given in M with the help of a description. We might perhaps say that ‘b’
is therefore only an apparent proper name, not a genuine one. And we
might try to correct the solution proposed by requiring that genuine
proper names be used, not those which are defined or interpreted by de-
scriptions. This attempt, however, would lead us into serious difficulties.
A moment’s reflection shows that most things have no proper names.
Some logicians — for example, Russell* and Quine* — do not accept indi-
vidual constants as primitive signs but only as abbreviations of compound
expressions. Thus the distinction between genuine and apparent proper
names of individuals is rather problematic. Even if there are genuine
proper names for some individuals, how should the extension of a descrip-
tion be given whose descriptum has no proper name? It is clear that the
attempted solution is inadequate in its present form.

However, I believe that another distinction will serve the purpose for
which the distinction between proper names and descriptions was in-
tended. To simplify the analysis let us take not a system like S„ whose
individual constants are names of things, but language systems of the
following kind. The individuals are p)ositions in an ordered domain.
Among the individual expressions there are some of a special kind, called

■ RusseH’s language contains names for qualities but not for particulais, i.e., individuals in
our sense (see [Inquiry], p. 117).

> Quine r^ards all individual constants as abbreviations for descriptions (see [M.L.], pp.

149 ff).

$ 18. L-DETERMINACY OP INDIVIDUAL EXPRESSIONS

expressions of standard form, which fulhl the following condition: (i) if
two expressions of standard form are given, then we can see from their
forms the positional relation between the two positions. Fof systems of a
simple structure (for example, the system Sj discussed in this section, in
contrast to the language of physics discussed in the next section) the fol-
lowing additional condition is fulfilled: (2) for every position, there is
exactly one expression of standard form. Languages of this kind may be
called coordinate languages, in distinction to name languages like S,.^
Let us take as an example a language system S3 in which the basic order
of the positions has the simple structure of a progression, a discrete linear
order with an initial position but no end. Let ‘o’ be taken as individual
constant for the initial position; if an individual expression of any form,
standard or not, is given as an expression for some position, an expression
for the next following position is formed from it by the adjunction of a
prime As individual expressions of standard form we take ‘o’, together
with those expressions consisting of ‘o’ followed by one or several primes.
Thus ‘o’, ‘o”, ‘o"’, ‘0'"’, are the standard expressions for the first four
positions.

Let S3 contain predicator signs for qualitative properties to be at-
tributed to the positions, say ‘B’ for the property Blue, ‘C’ for Cold, ‘S’
for Soft. Furthermore, S3 contains, like S„ the customary connectives,
individual variables with quantifiers, and individual descriptions. As
common descriptum for all descriptions which do not satisfy the unique-
ness condition, we take, of course, the initial position; hence ‘o’ takes the
place of ‘a*’ (see § 8). Thus, for example, the description ‘ Ux) (Bjc • Ct)’
means the same as ‘ the one position which is both blue and cold (or the
position o if no or several positions are both blue and cold)’. [As previous-
ly, we shall usually omit the phrase here included in parentheses.]

For the purpose of the subsequent examples, we presuppose this factual
assumption :

18-1. The second position (o') is the only one which is both blue and cold
and also the only one which is both blue and soft.

According to this assumption, the following holds:

18-2. ‘ (ix) (Bx • Cx) = o” is true (and, moreover, F-true).

18-3. ‘(ix)(Bx • Sx) = o” is true (and, moreover, F-true).

Suppose we ask the question, ‘What is the extension of the description
‘ (ix) (Bx • Cx) ’?’ because we do not know the facts (18-1) and wish to find

1 [Syntax], § 3. The system S, described in the text is similar to Language I dealt with in
[Syntax], Part I.

76 n. L-DETERMINACY

out which position is the descriptum. Let us consider the following an-
swers:

18-4. a. 'The extension of the description mentioned is the one position
which is both blue and cold.’

b. ‘The extension of the description mentioned is the same as that
of ‘(7Jc)(Ba: • S*)’.’

The answer i8-4a, although true, would certainly appear as unsatisfac-
tory; we should protest: ‘Yes, but which position is this?’. Sentence 4b is
likewise a true answer to our question, by virtue of the fact 18-1. It is not
so trivial an answer as 4a, but it still does not supply the information we
want. It does not tell us directly which position is the descriptum but
merely refers to this position by a qualitative characterization. After re-
ceiving the answer 4b, just as in the case of 4a, we still need factual obser-
vations concerning the qualities of the positions in order to discover which
position is the descriptum of the original description.

In contradistinction to those answers, each of the following two formu-
lations tells us actually what we want to know :

18-6. a. ‘The extension of the description is the second position’.

b. ‘The extension of the description is the same as that of ‘o'’.’

The same holds for 18-2. Each of these three answers supplies the informa-
tion directly. But there are other formulations which give the same infor-
mation in an indirect manner. In order to construct an example, let
‘ X . indicate a not too simple matrix in S3 without nonlogical con-
stants, which is fulfilled only by the position o'. [We may regard the indi-
viduEil expressions in S3 as expressions of natural numbers (‘o’ for Zero,
‘o'’ for One, etc.). Then we can introduce arithmetical symbols, for ex-
ample, ‘ > ’ for the relation Greater and ‘ X ’ for the function Product, re-
spectively. “ Let ‘ . X . indicate the matrix ‘ (x > o) • (2: X * = x)’,
which is satisfied only by the number One, hence by 0'.] Then the follow-
ing holds :

18-6. ‘(ix)(. . X . =0'’ is true (and, moreover, L-true).

(The sentence mentioned is L-true because it holds in all state-descrip-
tions, which differ only in the distribution of the qualitative properties.
The truth of the sentence can be shown by using only the semantical rules;
these include the rules determining the basic structure and the explicit
and recursive definitions involved.) Hence we obtain:

4 These and other arithmetical symbols can be introduced in a system like Sj with the help
of recursive definitions in the customary way (see, for instance, [Syntax], | 20).

77

§ 18. L-DETERMINACY OF INDIVIDUAL EXPRESSIONS

18 - 7 . The extension of ‘ (i*) is the same as that of ‘o".

Hence also the following holds, because of i8-5b:

18 - 8 . The extension of the original description ‘ (ix) (Bx • C*) ’ is the same
as that of . a; .

May we regard this statement i8-8 as a complete answer to our question?
It must be admitted that it characterizes the extension of the original de-
scription only in an indirect way; this it has in common with i8-4b. In
another respect, however, which is of a fundamental nature, i8-8 is dif-
ferent from that former answer and like those formulations which we re-
gard as complete answers, that is, iS-sa and b and 18-2. If we receive 18-8
as an answer, then, in order to derive from it the complete and direct
answer i8'5a or b, we need not make observations concerning the qualities
of the positions, as in the case of the answer i8-4b; all we have to do is to
carry out a certain logico-arithmetical procedure, namely, that which
leads to the result 18-6. Thus there is this fundamental difference: 18-6
states an L- truth, while 18-3 states an F-truth. The following two results
follow from the ones just mentioned (18-9 from 18-6, 18-10 from 18-3),
according to the definitions of L- and F-equivalence (3-5b and c) :

18 - 9 . ‘{ix)(. . X . and'o" are L-equi valent.

18 - 10 . ‘(’*) (Ba: • S*)’ and ‘0" are F-equivalent.

It is because of the L-equivalence stated in 18-9 that we also say that
18-8 actually gives the extension, although indirectly. Thus it becomes
clear that the difference between an answer giving the extension and one
merely describing it does not simply consist in the difference between the
use of a standard expression and that of a description. If a standard ex-
pression is used, the extension is certainly given ; but it may also be given
by a description, provided this description is L-equivalent to a standard
expression, as ‘ (iz) (. . x . is, according to 18-9. If, on the other hand, a
description is not L-equivalent to any standard expression, .then by using
it we do not give, but merely describe, the extension in question. Note that
every individual expression is an expression of exactly one position and
hence is equivalent to exactly one standard expression. Therefore, if an
expression is F-equivalent to some standard expression, as, for example,
‘(ia:)(Bx • Sx)’ is according to 18-10, then it cannot be L-equivalent to
any standard expression.

The results here found will help us in constructing, in the next section, a
definition for the L-determinacy of individual expressions.

78 n. L-DETERMINACY

§ 19. Definition of L-Determinocj of Individual Expressions

For a simple coordinate language like Sj (§ i8), we define as L-determinate
those individual expressions which are L-equivalent to standard expressions.
The problem of the definition of L-determinacy for more complex coordinate
languages, like the language of physics Sp, is briefly discussed. Finally, it is
shown how the concept of L-determinacy can be applied also to name languages
if the metalanguage is a coordinate language.

In the preceding section, we analyzed the individual expressions in the
system Sj, which was chosen as an example of a coordinate language of
simple structure. Analogous considerations hold for other systems in
which there are individual expressions of standard form which fulfil both
conditions (i) and (2), mentioned earlier. For the following definition of
Lf-determinacy it is presupposed that 5 is a system for which a standard
form has been determined which fulfils those conditions. This definition
is suggested by the results of our discussion in the preceding section.

19 - 1 . Definition. An individual expression in the system S is L-deter-
minate = m it is L-equivalent to an individual expression of standard form
in 5 . (This obviously includes the standard expressions themselves.)

That this definition satisfies our previous requirement, 17-4, is seen as
follows: If a given individual expression is L-equivalent to a standard ex-
pression, then those semantical rules on which this L-equivalence (in
other words, the L-truth of the corresponding = -sentence) is based suf-
fice to give its extension, namely, the position corresponding to the stand-
ard expression. On the other hand, if a given individual expression is not
L-equivalent to a standard expression, then it is, as we have seen, F-
equivalent to a standard expression. Therefore, in this case the semantical
rules do not suffice to give its extension; this can be given only by a factual
statement.

It should be noticed that there is, in general, no effective decision pro-
cedure for the concept of L-determinacy just defined. Still less is there a
general effective procedure for the evaluation of any given L-determinate
individual expression, that is, for its transformation into an L-equivalent
standard expression. (Joing back to the example of the system S3 with
arithmetical symbols (see the explanations preceding 18-6), —

o" X o")’ can be transformed into'o""’ simply by calculation, that is to
say, by repeated application of the recursive definitions. On the other
hand, the transformation of ‘ (ix) . x . i.e., ‘ (ix) [(x > o) • (x X x =

x)]’, into ‘o'’ requires the proof of a universal arithmetical theorem,
which states that every number except i lacks the describing property;
and it is clear that there cannot be a fixed effective procedure for finding

s 19. L-DETERMINACY OF INDIVTOUAL EXPRESSIONS 79

proofs of this kind. In cases like the two examples in S, just given, the
L-determinacy is easily established by the fact that both descriptions do
not contain any nonlogical constants. If, however, nonlogical constants
occur, then we have, in general, no effective procedure for deciding about
L-determinacy.

The basic order of the positions in a coordinate language 5 may be quite
different from the simple order in S^; but the procedure leading to a defini-
tion of L-determinacy will still be essentially the same. We first choose
among the individual expressions of the system those which we wish to re-
gard as of standard form. The choice is fundamentally a matter of conven-
tion, provided that, of the requirements stated earlier, at least the first is
fulfilled. The simplicity of the forms and the possibility of recognizing the
positional relations in a simple way will usually influence the choice. If the
primitive constants of the language system are divided into logical and
descriptive (i.e., nonlogical) constants (see § 21), then only expressions in
which all constants are logical will be taken as standard form.

As an example of a system with a different basic order, let us briefly
consider a coordinate language of physics Sv, leaving aside the technical
details. Here the individuals are space-time points within a coordinate
system chosen by convention. First, a standard form for expressions of
real numbers in Sp must be chosen. Here this is a much more complicated
task than in the case of natural numbers (as in Sj). The standard expres-
sions must enable us to find the location of positions and the distance be-
tween two positions with any desired degree of precision. This means that
for the representation of real numbers as systematic (e.g., decimal or dual)
fractions, we must have an effective procedure for computing any required
number of digits.* Since a space-time point is determined by three space
coordinates and one time coordinate, a standard individual expression in
Sp will consist of four standard real-number expressions.

A continuous coordinate language like Sp is; in certain respects, funda-

s This requirement can be stated in exact terms as follows. For every real number there is
a unique representation in the decimal system if we exclude decimals which, from a certain
place on, contain only the figure ‘g’. The integral part is a natural number; the fractional part
corresponds to a function f(») whose value gives the nth digit after the decimal point. (For
example, for ir = 3.1415 , f(i) — i, ffa) = 4, ffj) = i, ((4) = 5, etc.) If, then, a real-

number expression consists of an expression of its integral part (say, in the ordinary decimal
notation) and an expression for the function f corresponding to its fractional part, then this
real-number expression is computable if the expression for f is computable in the sense of
A. M. Turing (“On Computable Numbers”, Proc. London Math. Soc., Vol. XLII [1937]).
Turing has shown that this concept of the computability of a function coincides with Church’s
lambda-definability and with the concept of general recursiveness due to Herbrand and Gddel
and developed by Kleene (see Turing, “Computability and X-Definability”, Journal of Sym-
bolic Logic, Vol. II [1937]).

8o

n. L-DETERMINACY

mentally different from a discrete coordinate language like S3. The first
important difference consists in the fact that no language (with expres-
sions of finite length) can contain expressions for all real numbers.*' There-
fore, Sp cannot contain individual expressions for all individuals, that is,
space-time points — let alone individual expressions of standard form.
Thus here the second of the two conditions for standard expressions can-
not be fulfilled; only the first is required. Another difference is the follow-
ing: There is no general effective method which would enable us to decide
for any two standard individual expressions whether or not they are
equivalent, that is, refer to the same position — in other words, whether
or not their (four-dimensional) distance is o. However, if two standard ex-
pressions are given, we can determine their distance in the form of a com-
putable function. Hence, for any positive rational number S, no matter
how small it may be chosen, we can establish either that the distance is ^5
and hence that the positions are distinct, or that the distance is g6, that
is, the positions are either identical or certainly not farther apart than 5.

We cannot here go any further into the technical details of the problem
of L-determinacy for the individual expressions in Sp. The problems
which ought to be investigated are the following. It is clear that not all
individual expressions in Sp can be equivalent to standard expressions.
The question should be examined as to whether the standard form can be
chosen in such a manner that at least all those individual expressions
which do not contain nonlogical constants are equivalent (and hence
L-equivalent) to standard expressions. If so, L-determinacy can be de-
fined for Sp as in 19-1. Otherwise, a more complicated definition will per-
haps be necessary; but it will, in any case, be such that L-equivalence to
a standard expression is a sufficient, though perhaps not a necessary, con-
dition for L-determinacy.

So far we have applied the concept of L-de terminate individual expres-
sions only to coordinate languages. Now let us consider name languages,
as, for example, S,. In a language of this kind we have no individual ex-
pressions which exhibit their positional relations directly by their form.
We may have individual expressions in the form of descriptions using
qualitative describing properties; furthermore, there may be individual
constants which are either primitive or perhaps introduced by definitions
as abbreviations of descriptions. However, even a primitive individual
constant in a name language 5 may, under certain conditions, be L-
determinate if the metalanguage M is a coordinate language. For every
primitive individual constant in 5 there is a rule of designation in M

‘ See [Syntax), § 6od.

§ 20. L-DETERMINACY OF PREDICATORS 8i

which tells us which individual is meant by the expression. This rule refers ^
to the individual by an individual expression in JIf . Now if J/ is a coordi-
nate language and the individual expression used in the rule is L-determi-
nate in Af in the sense earlier explained for coordinate languages, then we
may likewise regard the individual constant in S as L-determinate. This
extended use of the term ‘L-determinate’ seems natural, since it satisfies
our earlier requirement 17-4: The semantical rules give the extension of
the constant, that is, the location of the position to which the constant
refers. This may be illustrated by the following example. Suppose the
expressions ‘o’, ‘o'’, ‘o"’, etc., occur, not in the object language S, which
is supposed to be a name language with individual constants ‘a’, ‘b’, etc.,
but in M, and that they refer, as explained earlier for Sj, to the positions
in a discrete linear order. Suppose, further, that the following two rules
are among the semantical rules of 5 formulated in M; they are rules of
designation for the primitive constants ‘a’ and ‘b’ :

1&-2. a. ‘a’ designates the position o".

b. ‘b’ designates either the one position which is both blue and cold,
or the position o if no or several positions are blue and cold.

We would in this case construct the definition of L-determinacy in such a
way that ‘a’ will be called L-determinate but ‘b’ not. (We omit here the
actual construction.) These results will then be in agreement with the re-
quirement 17-4. We see from rule i9-2a that the extension of ‘a’ is the
third position. On the other hand, the semantical rules do not give the
extension of ‘b’ but merely describe it (in rule 19- 2b) ; it can be given only
by the addition of a factual statement to the rules. Thus the first part of
the factual statement 18-1, together with the rule i9-2b, tells us that the
extension of ‘b’ is the second position (o').

§ 20. L-Determinacy of Predicators

A predicator (in a coordinate language like Sj) is said to be L-determinate
if every full sentence of it with individual expressions of standard form is
L-determinate. This holds if the intension of the predicator is a positional or
mathematical, rather than a qualitative, property. The analogous definition
for functors is briefly indicated.

The concept of the extension of a predicator, especially if we consider
predicators of degree one, seems entirely clear and unproblematic. For
example, the extension of the predicator ‘H’ in the system S, is the class
Human because its intension is the property Human. We began the ex-
planation of the method of extension and intension with the customary
and apparently clear and simple distinction between classes and proper-

83

n. L-DETERMINACY

ties (§ 4). We took this distinction as a model and framed the distinctions
between the extension and the intension of sentences and of individual ex-
pressions in analogy to it (§§ 6 and 9). A closer inspection shows, however,
that a serious difficulty is involved even in the concept of the extension of
a predicator. We could leave this difficulty aside in our earlier discus-
sions, but for our present purpose we have to face it and try to overcome
it. In order to find an adequate definition for L-determinacy of predicators
we have to make clear the means by which a class can be given. We shall
see presently that this problem cannot be solved without first solving the
problem of the way in which the extension of an individual expression can
be given. This was our reason for first discussing individual expressions in
the two preceding sections.

Suppose we ask somebody for information about the extension of the
membership of Club C ; that means that we want to learn who is a member
of C and who is not. The answer ‘the extension is the class of the members
of C’ is, although true, entirely trivial and hence would not satisfy us. Nor
would an answer like ‘the class of those boys in this town who either are
between fifteen and sixteen years old or have red hair’. Although this
answer is not trivial, it still does not give the extension but merely de-
scribes it with the help of another complex property which happens to
have the same extension. What we want is not an indirect characteriza-
tion of the membership by an intension but a membership list. Would
every kind of membership list satisfy us? We see easily that some kinds
would not. Thus the problem arises: What kind of membership list does
actually give the extension? Suppose that we are given a statement which
lists all the members of the club but does so by formulations like these :
‘the eldest son of Mr. Jones’, ‘the boy friend of Mary’, etc. We should
again reject this statement, although it enumerates aU members, because
it does so by descriptions. Thus we see that a certain class is not merely
described but actually given by a statement if this statement (i) refers
to each of the members of the class and (2) does so by the use of individual
expressions, which, in turn, do not merely describe but give the indi-
viduals — in other words, by the use of L-determinate individual expres-
sions. This shows that the concept of L-determinacy of predicators pre-
supposes the concept of L-determinacy of individual expressions.

We presuppose for the following discussions that 5 is a coordinate lan-
guage of a simple structure similar to S3, as explained in the beginning of
§ 19 ; that a standard form of individual expressions has been defined for
S; and that L-determinacy of individual expressions in 5 is defined by
our previous definition (19-1).

s 20. L-DETERMINACY OF PREDICATORS 83

The condition fonnulated above for a statement giving a class is suf-
ficient but not necessary. The statement need not give an enumeration of
all members of the class; if this were necessary, then only finite classes
could be given. It is sufficient and also necessary that the statement
logically imply the truth of all those true singular sentences in 5 which
say of an individual that it is or that it is not a member of the class, where
the individual expressions occurring are L-determinate.

It would even be sufficient to require this merely for all the individual
expressions of standard form in S; it is easily seen that it also holds, then,
for all L-determinate individual expressions because they are L-equivalent
to standard expressions, according to the definition 19-1.

In order to give examples let us go back to the coordinate language S3
with ‘o’, ‘o'’, ‘o"’, etc., as standard expressions. Suppose that the state-
ment ‘the positions o and o'" and no others are blue’ is true. Then it
gives the extension of the predicator ‘B’, because from this statement, to-
gether with the semantical rules, we can infer that ‘B(o)’ and ‘B(o'")’
are true, while all other full sentences of ‘B’ with a standard expression
are false. Let us introduce into S3 the customary notation ‘ ’

for a finite class indicated by an enumeration of its members; the defini-
tion can be written with the help of a lambda-operator as follows:

20-1. Abbreviation, . . . x„]’ for ‘(Xy)[(y = *,) V (y = V . . . V

(y - O]’.

Then the extension of ‘B’ in the above example can be given also by this
statement: ‘the extension of ‘B’ is the same as that of ‘{o, 0'")’.

These considerations suggest the following definition for L-determi-
nacy of predicators in a system S (of the kind indicated above) . It presup-
poses the definition of L-determinacy for sentences {2-2A).

20 - 2 . Definition. A predicator in 5 is L-determinate =Df every full sen-
tence of it with individual expressions of standard form is L-determinate.

We see easily that this definition fulfils our earlier requirement 17-4;
the concept defined applies if and only if the semantical rules alone, with-
out any factual knowledge, suffice to give the extension of the predicator
in the sense explained above, because a sentence is L-determinate if and
only if the semantical rules suffice to determine its truth-value (conven-
tion 2-l).

We see that any predicator in S3 of the form '{\x) (. . x . where any
molecular combination of ‘= ’-matrices with ‘x’ and standard expressions
stands in the place of ‘. . x . .’, is L-determinate. Therefore, ‘|o, o"'}’ is
L-determinate, and likewise any other predicator of the form ‘{. . .1’

84

II. L-DETER3nNACY

where all individual expressions occurring are of standard form. Let us
define in the customary way the signs ‘A’ and ‘V’ of the null class and
the universal class, respectively, or, more exactly, of the L-empty prop-
erty and the L-universal property, respectively :

20-3. Abbreviations.

a. ‘A’ for ‘(^J?)[~(x = x)]’.

■ b. ‘V’for‘(Xa:)[x = x]’.

We see immediately that the two predicators here defined are L-determi-
nate, because all full sentences of ‘A’ are L-false and all of ‘V’ are L-true.
But there are other, more complicated predicators which likewise are
L-determinate, among them all lambda-expressions with any purely
arithmetic conditions. Take, as an example, the predicator ‘(Xx;)[Prime
(*)]’, where ‘Prime’ is defined so that it holds for all prime numbers (that
means, for all positions with a prime coordinate).’ This example shows
that even a predicator whose extension is infinite and therefore cannot be
given by an enumeration may be L-determinate. This is the case if the in-
tension is of a mathematical, rather than of an empirical, nature; in other
words, if the intension is a positional, rather than a qualitative, property.
That, for instance, the position o'", corresponding to the number Three,
belongs to the extension of ‘Prime’ is found by a purely logico-mathe-
matical procedure, that is, a procedure based upon the semantical rules
and not involving the qualitative properties of that or any other position.
On the other hand, for establishing that the position o'" belongs to the
extension of ‘B’, we need not only the semantical rules but, in addition,
an observation yielding the result that this position has the color Blue.

Here, again, for the concept of L-determinate predicators there is no
effective method of decision, since there is none for the concept of L-de-
terminate sentences on which it is based. For example, let x be called a
Fermat exponent if a: > 2 and if there are positive integers «, v, and w
such that ^’'-1-1'* = Tsf. A predicator for this property, say ‘Fer’, can
easily be defined in S3. ‘Fer’ is an L-determinate predicator because every
full sentence of it with a standard individual expression is an L-determi-
nate sentence. For most of these sentences it is at present unknown
whether they are true or false, and there is no decision method for deter-
mining their truth-value. Nevertheless, they are L-determinate, because
their truth-values are independent of colors or any other qualitative
properties of the corresponding positions. For the number Three and some
others it is known that they are not Fermat exponents. This has been

' Arithmetical concepts of this kind can be defined in a language similar to S3 with the help
of recursive definitions (see, for example, [Syntax], § 20).

8s

§ 21 . LOGICAL AND DESCRIPTIVE SIGNS

shown by a mathematical proof; thus the result is independent of the
qualitative properties of the positions. Therefore, the sentence ‘~Fer(o'")’
holds in every state-description and hence is L-true in Sj.

It may be remarked incidentally that a definition of L-determinacy for
functors and compound functor expressions can be given which is quite
analogous to that for predicators (20-2). Here it would likewise be required
that every full sentence in which the argument expressions and the value
expression are of standard form be L-determinate. Thus, all signs or ex-
pressions for arithmetical functions are L-determinate. For example, the
functor' in S3 is L-determinate because every full sentence with stand-
ard expressions is L-determinate; for instance, ‘o' -f- o" = 0'"’ is L-true.
On the other hand, a functor for a physical magnitude, for example, tem-
perature (say, in the language of physics, Sp) is not L-determinate, because
a sentence saying that the temperature at a certain space-time point has a
certain value is not L-determinate.

§ 21 . Logical and Descriptive Signs

We make use in this section of the customary distinction between logical and
descriptive (nonlogical) signs. For the system S3 (restricted to primitive signs)
the classification is simple: the primitive predicates are descriptive, all other
signs are logical. If a designator in S3 contains only logical signs, then it is L-de-
terminate. A designator in S3 is L-determinate if and only if it is L-equivalent
to a designator containing only logical signs. This could be taken as an alterna-
tive way of defining L-determinacy.

In this section we make the customary distinction between logical and
descriptive, i.e., nonlogical signs.* With its help we shall then make a cor-
responding distinction for expressions, which is especially important for
designators. Then we shall investigate the relation between this distinc-
tion and the distinction between L-determinate and L-indeterminate
designators. The concepts of logical and descriptive signs will seldom be
used in the rest of the book.

We shall define the concepts mentioned for two example, systems, one
a coordinate language and the other a name language. As coordinate
language we take the system S3 discussed in the preceding sections; it con-
tains ‘o’, ‘o'’, etc., as individual expressions of standard form. As name
language we take a system S,' which is like our system S, with this excep-
tion: We suppose that the individual constants in it, say ‘a*", ‘a’, ‘b’, etc.,
are interpreted by the semantical rules of S,' as referring, not to things,
as in S, (see rule i-i), but to positions in an ordered domain (as, for ex-

* For more detailed explanations see [I], § 13.

86

II. L-DETERMINACY

ample, in rule 19- 2a). Therefore, these constants are L-determinate, as
explained earlier (at the end of § 19). Both systems are here supposed to
contain only primitive signs, not defined signs. The predicates in both sys-
tems are supposed to be interpreted by the semantical rules as designating
qualitative properties or relations like Blue, Cold, Colder, and the like
(as explained for S3 in § 18).

The distinction between logical and descriptive signs of the systems Sj
and S,' is made in the following way by enumeration of particular signs and
kinds of signs.

21-1. The following signs are regarded as logical:

a. The individual variables.

b. The connectives; the operator signs ‘ 3 ’, ‘1’, ‘ X’; the parentheses.

c. In S,', the individual constants; in S3, ‘o’ and
21-2. The predicates are regarded as descriptive signs.

The corresponding distinction for expressions is now defined in 21-3 ; to
be descriptive is taken, so to speak, as a dominant property; to be logical
as a recessive property.

21-3. Definitions.

a. An expression is logical =dj it contains only logical signs.

b. An expression is descriptive =Df it contains at least one descrip-
tive sign.

Thus the standard expressions ‘o’, ‘o'’, etc., in S3 are regarded as logical.
This seems justified because they refer here not to things but to positions
in a basic, presupposed order. We may even interpret them as referring
to pure numbers. In a word translation of ‘B(o"')’ the expression ‘0'"’
corresponds in this interpretation to the italicized part in ‘the position
correlated to the number Three is blue’, while the predicate ‘ B’ corresponds
to the whole nonitalicized part of this sentence.’ This interpretation is just
as adequate as the ordinary interpretation by ‘ the position correlated to the
number Three is blue’. We might even say that these are merely two differ-
ent formulations for the same interpretation, since the translation of the
whole sentence is the same in both cases, and hence the truth-condition
of the sentence remains likewise the same.

In addition to the individual expressions of standard form in S3 (e.g.,
‘0'"’) and in S( (here we take the individual constants as standard form),
both systems contain individual descriptions.

The following results concern the system S3. They hold likewise for S,',

* This interpretation has, furthermore, the advantage that a sentence which says that the
universe of individuals is infinite is not factual but L-true. Thus the difficulty usually connected
with the so-called Axiom of Infinity is here avoided (see [Syntax], p. 141).

§ 21. LOGICAL AND DESCRIPTIVE SIGNS

87

provided that the basic order of its universe of individuals is either the
same as in Sj or has a similar simple structure and provided that the rules
of designation formulated in M for the individual constants in S' use only
individual expressions of standard form; this standard form in M may,
for example, be the same as in S3.

21-4. Every sentence in S3 which contains only logical signs is either L-
true or L-false; and there is an effective decision method for determining
which of the two is the case.

21-6. Every (closed) description in S, is L-determinate; and there is an
effective procedure for transforming it into an individual expression of
standard form.

21-6. Every closed lambda-expression in Sj is L-determinate; and there
is a decision method for any full sentence of the lambda-expression with
any individual expression of standard form.

The proofs of these theorems and the decision methods mentioned can-
not be given here, but they are rather simple.'" They are based on the fol-
lowing circumstances: (i) since no predicates occur, the ultimate compo-
nents are s-matrices; (2) an ^-sentence with two standard expressions is
L-true if the two standard expressions are alike, and otherwise it is
L-false.

The three results can be combined into one as follows :

21-7. Every designator in Sj which contains only logical signs is L-
determinate.

There are, however, also L-determinate designators which contain de-
scriptive signs. For example, ‘P(o) V ~P(o)’ is L-true; ‘ {\x) (Pae V ~Px)’
is L-universal, and hence L-equivalent to ‘V’ (2o-3b); and ‘(ix)
(Pjc V ~Pii:) ’ is L-equivalent to ‘o’; thus these three designators are all
L-determinate.

It follows from 21-7 that any designator L-ecfuivalent to one containing
only logical signs is likewise L-determinate. Now it can be shown that the
converse of this holds too. (i) If a sentence is L-determinate, then it is
either L-true or L-false; therefore, it is L-equivalent either to ‘o = o’
or to the negation of this sentence, (ii) If a description is L-determinate,
it is L-equivalent to a standard expression, according to the definition
19- 1, (iii) It can be shown that, if a closed lambda-expression in S3 is
L-determinate, either its extension or the complement of its extension is
finite; therefore, the lambda-expression is L-equivalent to one of the form
‘ (Xa;) (. . a; . whose scope is constructed with the help of connectives

For further details see [Modalities], $$ ii and 12, especially Ti2-2f.

88

II. L-DETERMINACY

out of = -matrices with and standard expressions. Thus the following
holds;

21 - 8 . A designator in Sj is L-determinate if and only if it is L-equivalent
to one containing only logical signs.

For S3 and similar systems, L-determinacy for designators could be
generally defined by the sufficient and necessary condition stated in 21-8.
This alternative method presupposes only the concepts of logical signs
(21-1) and of L-equivalence of designators (3-5b), hence of L-truth of
sentences (2-2); it would replace the three separate definitions of L-
determinacy for sentences, individual expressions, and predicators earlier
given (2-3d, 19-1, 20-2).

Now we can easily see that if two designators in S3 which contain only
logical signs are equivalent, then they are L-equivalent. Since they are
equivalent, the =-sentence containing them as components is true (3-5a)
and therefore L-true, according to 21-4; hence they are L-equivalent
(3-Sb). From this result the following more general theorem can be de-
rived with the help of 21-8 and the transitivity of equivalence and L-
equivalence:

21 - 9 . If two L-determinate designators in S3 are equivalent, then they are
L-equivalent.

§ 22 . L- Determinate Intensions

If a designator is L-determinate, then all designators L-equivalent to it are
likewise L-determinate. We shall say of the common intension of these designa-
tors that it is an L-determinale intension. For any extension, there are, in gen-
eral, many corresponding intensions; but there is among them exactly one
L-detenninate intension.

The results which will be stated here can be proved in an exact way for
the system S3. But it can be shown in an informal way that they hold like-
wise for any system S, provided the concepts of L-truth and L-determi-
nacy are defined for 5 in such a manner that our requirements for these
two concepts (2-1 and 17-4, respectively) are fulfilled. In the following
discussion it is presupposed that these requirements are fulfilled.

22 - 1 . If two L-determinate designators in 5 are equivalent, then they are
L-equivalent.

Applied to S3, this is the same as 21-9, which was proved with the help
of the distinction between logical and descriptive signs. The general theo-
rem for a system 5 can be seen to hold in the following way, which does
not presuppose such a distinction. Since the two designators are equiva-

§ 22. L-DETERMINATE INTENSIONS

89

lent, they have the same extension (5-1). Since they are L-determinate,
the semantical rules suffice for establishing that both have this same
extension (17-4) and hence that they are equivalent (5-1) and hence that
their s-sentence is true (3-sa) ; therefore, this s-sentence is L-true (2-1) ;
hence the two designators are L-equivalent (3-5b).

22 - 2 . If a designator in S is L-equivalent to an L-determinate designator,
then it is itself L-determinate.

For S3, this follows from 21-8 because of the transitivity of L-equiva-
lence. That it holds generally for 5 is seen as follows; If the condition in
22-2 is fulfilled, the semantical rules suffice for establishing the extension
of the second designator and the identity of extension for the two desig-
nators, and thereby the extension of the first designator.

Suppose an L-determinate designator in 5 is given. It possesses a certain
intension. Any other designator having this same intension is L-equiva-
lent to the first and hence likewise L-determinate, according to 22-2. Let
us call an intension of this kind an L-determinate intension. Thus,
roughly speaking, an L-determinate intension is such that it conveys to
us its extension. For every extension, there are, in general, many cor-
responding intensions; but among them there is exactly one L-determinate
intension, which may, in a way, be regarded as the representative of this
extension (not, of course, in the sense in which a designator may be said
to represent, or refer to, its extension). This one-one correlation between
extensions and L-determinate intensions will become clearer with some
examples.

For sentences, there are only two extensions, the two truth-values.
Truth and Falsity. There are many L-determinate sentences whose ex-
tension is the truth-value Truth, namely, all the L-true sentences, e.g.,
‘Pa V ^Pa’ (in S,). Since they are L-equivalent to each other, they have
the same intension, namely, the L-true or necessary proposition. Thus
this proposition is the one L-determinate intension corresponding to the
extension Truth. Analogously, the L-false or impossible proposition is the
L-determinate intension which corresponds to the extension Falsity. For
predicators, there are infinitely many extensions, namely, classes of indi-
viduals. If, as in S, and S3, the number of individuals is denumerably infi-
nite, the number of classes of individuals is nondenumerable; since the
number of (finite) expressions in any language system 5 is, at most, de-
numerable, not all classes of individuals can be extensions of predicators
in 5 . For an extension referred to by a predicator in S there is not neces-
sarily always a corresponding L-determinate intension expressed by a

90

II. L-DETERMINACY

predicator in S, because not every predicator has an equivalent L-determi-
nate predicator. Whether a certain L-determinate intension is or is not
expressed by a predicator in 5 depends on the means of expression in 5 .
The L-determinate intension corresponding to the null class of individuals
is the L-empty property; in Si and S3 this intention is expressed, for ex-
ample, by ‘(Xx)[~(a: = x)]\ The L-determinate intension corresponding
to the universal class is the L-universal property, expressed by ‘(Xa:)
[x = x]’. The L-determinate intension corresponding to the class whose
only members are the positions o, o", and o'" is the property of being one
of these three positions, which is expressed in S3 by ‘(Xa:)[(a; s o) V
(x s o") V (x = 0'")]’. On the other hand, suppose that the primitive
signs of S3, mentioned earlier, are the only signs in S, and that S' is con-
structed from S3 by the addition of some recursively defined functors and
predicators, among them the predicator ‘Prime' for the property Prime
Number. Suppose, further, that all prime number positions, and only
these, happen to be blue. Then the extension of ‘B’ is the class of prime
number positions, and the corresponding L-determinate intension is the
property of being a prime number position. This intension is expressed in
S' by the L-determinate predicator ‘Prime’; but in S3 it is not expressed
by any predicator.

The extensions of individual expressions are the individuals, which in
S3 are the positions. For example, the extension of the description ‘ (ix)
(Bx • Cx) ’ in our earlier example is the second position (i.e., the position
next to the initial position, i8-5a). Therefore, the corresponding L-deter-
minate intension is the individual concept The Second Position, which is
expressed in Sj by the L-determinate individual expression ‘o'’. Generally
speaking, for every individual in S3 there is one L-determinate intension,
namely, the individual concept of that position ; this intension is expressed
in S3 by at least one L-determinate individual expression, for instance, by
the standard expression (‘o’, ‘o'’, etc.).

§ 23. Reduction of Extensions to Intensions

The one-one correlation between extensions and L-determinate intensions
suggests the identification of extensiorfs with the corresponding L-determinate
intensions. According to this method, which is discussed in this section but will
not be used in the remainder of the book, a class is construed as a positional
property. This leads to explicit definitions of classes, in distinction to the con-
textual definitions used by Whitehead and Russell.

The method of extension and intension introduced in the first chapter
assigns to every designator an extension and an intension. Thus our

§ 23 . REDUCTION OF EXTENSIONS TO INTENSIONS 91

semantical analysis of the designators seems to assume two kinds of
entities — extension and intensions. It has been mentioned earlier that
this assumption is not actually made, that, in fact, we merely use two
forms of speech which can ultimately be reduced to one. There are several
possibilities for this reduction; they fall chiefly into three kinds: (i) the
extensions are reduced to intensions; (ii) the intensions are reduced to
extensions; (iii) both extensions and intensions are reduced to entities,
which are, so to speak, neutral. We shall later explain several methods
of the first kind. The chief requirement that such a method must fulfil is
obviously this: two different but equivalent intensions must determine the
same extension. The methods of this kind to be explained later (§ 33,
methods (2) and (3)) give, not an explicit definition, but only a contextual
one. That is to say, a phrase like ‘the class Blue’ is not itself translated
into a phrase in terms of properties; instead, a rule is given for transform-
ing any sentence containing the phrase ‘ the class Blue’ into a sentence
referring only to properties.

Now the introduction of the concept of L-determinate intension (in the
preceding section) makes it possible to define extensions in terms of in-
tensions. This method requires that the universe of individuals in question
exhibit a basic order so that the concept of L-determinacy may be ap-
plied. It is not required that the object language be a coordinate language;
the basic order need not be exhibited by the individual expressions of the
object language; it is sufficient that it be expressible in the metalanguage.
We suppose for the following definitions, as we did in the preceding sec-
tion, that the concepts of L-truth and L-determinacy are defined for the
system 5 in such a manner that our requirements for these two concepts
(2-1 and 17-4) are fulfilled. The advantage of the method to be applied
here is that it supplies explicit definitions. It is based on the following
three results, which we found earlier: (i) to every intension there cor-
responds exactly one L-determinate intension ; -(ii) the L-determinate in-
tensions corresponding to any two intensions which are equivalent and
hence have the same extension are identical; (iii) therefore, there is a one-
one correlation between extensions and L-determinate intensions.

The method to be proposed consists simply in identifying extensions
with the corresponding L-determinate intensions.

23 - 1 . Definition. The extension of a designator in 5 = di the one L-determi-
nate intension which is equivalent to the intension of the designator.

The concept of the equivalence of intensions used in this definition was
introduced (definition 5-3) with the help of the concept of the equivalence

91

II. L-DETERMINACY

of designators; the latter concept was defined (3-sa) by the truth of an
= -sentence and hence does not presuppose the concept of extension.

Our principal requirement for extensions was that they be identical for
equivalent designators (5-1). This requirement is fulfilled by the present
definition 23-1 (see (ii) above).

Although we have usually spoken of intensions only as intensions of
designators, occasionally reference was made to intensions independent
of the question of whether or not they were expressed by designators in the
system under discussion. Therefore, it may be useful to have the following
definition for the extension of (or, corresponding to, determined by) an
intension; here no reference is made to designators.

23-2. Definition. Tht extension of a given intension =Df the oneL-determi-
nate intension which is equivalent to the given intension.

Let us apply these definitions to the examples in the system S3 given in
the preceding section. Let us begin with predicators, because in this case
the concept of extension, that is, of class, is more familiar than in the
other cases. Classes are now identified with L-determinate properties, that
is, positional properties. Let us assume, for example, that the positions
o, o", o'", and no others, are blue. On the basis of this assumption, the
extension of the predicator ‘B’ in S3 is, according to the definition 23-1,
the intension of ‘(Xx:)[(2: = o) V (a: s 0") V (* = 0'")]’, that is, the
property of a position of being either o or o" or o'". And we say likewise,
according to the definition 23-2, that the extension of the property Blue
is the positional property just mentioned. However, it should be noted
that these two results are factual statements based on the factual assump-
tion mentioned. Our definitions do by no means say that the phrases ‘ the
extension of ‘B’ ’ and ‘the extension of the property Blue’, to which we
may add the third synonymous phrase, ‘ the class Blue’, mean the same
as ‘ the property of being either o or o" or 0'"’. The latter phrase is merely
equivalent to each of the three former phrases. What the definition 23-1
actually says is that the phrase ‘the extension of ‘B’ ’ means the same as
‘the L-determinate intension which is equivalent to the intension of
‘B’ ’ — in other words, ‘the positional property which is equivalent to the
(qualitative) property Blue’. It is a matter of fact, not of logic, that the
positional property which is equivalent to the property Blue is the
property of being either o or o" or o'".

Let us assume, further, that no position is both blue and cold. Then
the extension of ‘B *0’ in S3 is the null class; this is now identified
with the L-empty property, which is expressed in S3 by the predicator

$ 23. REDUCTION OF EXTENSIONS TO INTENSIONS

93

‘(X*)[ '^(x s x)]’. Suppose that all prime number positions, and no others,
are blue. Then the extension of ‘B* is the dass of prime number positions.
This class is now identified \rith the property Prime Number Position.

It may perhaps at first seem somewhat strange to regard classes not as
distinct entities corresponding somehow to properties but as properties of
a spedal kind. But a consideration of the examples given will remove or
mitigate the feeling of strangeness. For example, it might not seem very
unnatural to regard the intension of ‘(Xa:)[(x = o)\/ {x — o") V (* =
o'")]’ as a class when we consider the fact that this intension, in contrast
to L-indeterminate intensions, provides by itself an answer to the question
as to the individuals to which it applies and those to which it does not.

Now we are going to apply our definitions to sentences. If we approach
the matter naively, without careful analysis as to the nature of the
entities, we might perhaps be inclined to say that we know, at least
roughly, what we mean by the extension of a predicator (of degree one),
that is, a dass. However, if it is said that the extension of a sentence is a
truth-value, it is not at all dear what entities should be regarded as truth-
values. In our earlier discussion (in § 6), we left aside the diflhculty here
involved ; but now let us examine it and try to solve it. We consider here
languages which speak about extra-linguistic individuals, either physical
things with physical properties, as in S,, or positions, as in Sj, with physi-
cal properties (e.g., ‘the second position is cold’). Both the intensions and
the extensions of predicators are clearly extra-linguistic entities; both
properties of individuals and classes of individuals (no matter whether
regarded in the customary way or, according to the method here proposed,
as properties of a special kind) have to do with the individuals, not with
expressions in the language. The same holds for extensions and intensions
of individual expressions ; both individuals and individual concepts, what-
ever their specific nature may be, are certainly extra-linguistic entities.
Therefore, it seems natural to expect, by analogy, that intensions and
extensions of designators of all kinds are extra-linguistic entities. This
holds also for the intensions of sentences, the propositions. But what about
their extensions? What kind of entities are the truth-values which we
take as the extensions of sentences? We might perhaps be inclined to
answer that the truth-values are truth and falsity and that these two
terms are to be understood in their semantical sense. However, truth in
the semantical sense is a certain property of sentences, hence a linguistic
entity. [This does not imply that truth is a merely linguistic matter; truth
is dependent upon extra-linguistic facts; therefore, its definition must refer

94

II. L-DETERMINACY

to extra-linguistic entities. However, we are here not concerned with the
question of the entities to which the definition refers, but rather with the
question of the kind (logical type) of entity to which the concept of truth
belongs. And here the answer is: It is a property of sentences.] Therefore,
truth and falsity fall outside the domain to which all other intensions and
extensions belong. Now there is nothing in the situation that compels us
to take (semantical) truth and falsity as the extensions of sentences. All
that is required is that the extension of all true sentences be the same
entity and that the extension of all false sentences be the same entity
but something different from the first. There are obviously many different
possibilities of choosing in a not too arbitrary manner two extra-linguistic
entities such that the one is connected in a simple way with all true sen-
tences and the other with all false sentences. What type of nonlinguistic
entities should we choose? It seems most natural to choose either two
properties of propositions or two propositions. Let us consider some possi-
bilities of these two kinds. The most natural properties of propositions to
be considered would obviously be truth and falsity of propositions. [In
distinction to truth or falsity of sentences, these two concepts are not
semantical but independent of language.” Their relation to the semantical
concepts of truth and falsity is the same as the relation of the equivalence
of intensions to the equivalence of designators; see the definition 5-3 and
the explanations preceding it, including the footnote. They are singulary,
truth-functional connections.”] It would be simpler to take two proposi-
tions. We might, for example, take, on the one hand, the proposition
expressed by the class of all true sentences in S, and, on the other hand, the
negation of p^. [In systems like S, and S3, where we have state-descrip-
tions (§ 2), the proposition is expressed in a simpler way by the one
true state-description.] This device might perhaps appeal to those philos-
ophers who regard truth as involving in some sense the whole universe.'^
While this method takes two factual (contingent) propositions as exten-
sions, our own method (23-1) takes the two L-determinate propositions.
Here the extension of any true sentence is the L-true (necessary) proposi-

“ In the terminology of [I], they are absolute concepts; for their definitions, see [I], D17-1
and Di7-3.

“ ‘True’ in this sense is a connective with thecharacteristic TF and hence is redundant (e.g.,
‘(the proposition) that Scott is human is true’ and ‘Scott is human’ are L-equivalent sentences
in M); ‘False’ has the characteristic FT and hence is a sign of negation (compare [II], § 10).

‘J Lewis ([Meaning], p. 242) maintains a similar conception. The denotation or extension of
a proposition “is not that limited state of affairs which the proposition refers to, but the kind
of total state of affairs we call a world. . . . All true propositions have the same extension, name-
ly, this actual world; and all false propositions have the same extension, namely, zero-exten-
sion.”

§ 23. REDUCTION OF EXTENSIONS TO INTENSIONS 95

tion; and the extension of any false sentence is the L-false (impossible)
proposition. Here, likewise, we probably feel, at first, some reluctance to
regard propositions as truth-values or extensions. However, the connec-
tion between the two L-determinate propositions and what we usually
regard as the truth-values is so close and natural that it is perhaps not too
artificial to take these propositions as extensions of sentences.

Now let us apply the new method to individual expressions. Let us again
assume that only the second position o' in S3 is both blue and cold. We
said earlier that, on the basis of this assumption, the extension of ‘ (1*)
(Bx • Cx) ’ is the second position. We say now, instead, that the exten-
sion of this description is the individual concept The Second Position. In a
sense this may be regarded as merely a change in formulation. We may
even use the same formulation as before, by saying; “The extension of the
description'is o'”. The change appears only when we add to ‘o” a specify-
ing noun. But this addition serves merely for greater clarity. The new
method does not lead to the result ‘ the extension is not the individual
(or position) o”. The situation is, rather, this: the new method in its pri-
mary formulation does not use the terms ‘ individual’, ‘ class’, ‘ truth-value’
at all; thus ‘o” and ‘the individual concept o” are synonymous. In a
secondary formulation those terms might be reintroduced under the new
method, in analogy to the introduction of ‘extension’ by 23-1 and 23-2.
But then again a combination of any of these three terms with an L-deter-
minate designator is synonymous with the designator alone. Thus, for
example, on this method the phrases ‘ the individual (or position) o”,
‘o", and ‘the individual concept o'’ all mean the same. Likewise, if ‘A’ is
used in M, the phrases ‘the class A’ (or ‘the null class’), ‘A’, and ‘the
property A ’ all mean the same.

I will not decide here the question of whether the method of taking
L-determinate intensions as extensions is or is not natural. It may suffice
to have shown that this method meets the formal requirements of a solu-
tion to the problem of extensions. For the further discussions in this book,
this method will not be presupposed; most of the discussions will be inde-
pendent of any particular specification of the nature of the entities chosen
as extensions, beyond the general requirement that equivalent designators
have the same extension (5-1).

CHAPTER III

THE METHOD OF THE NAME-RELATION

The method of the name-relation is an alternative method of semantical
analysis, more customary than the method of extension and intension. It con-
sists in regarding expressions as names of (concrete or abstract) entities in ac-
cordance with the folloa’ing principles (§ 24): (i) every name has exactly one
nominatum (i.e., entity named by it); (2) any sentence speaks about the
nominata of the names occurring in it; (3) if a name occurring in a true sentence
is replaced by another name with the same nominatum., the sentence remains
true. An examination of the method shows that its basic concept involves an
essential ambiguity (§25) and that it leads to an unnecessary duplication of
expressions in the object language (§§ 26, 27). The most serious disadvantage
of the method consists in the fact that the third of the principles mentioned,
although it seems quite plausible, leads in certain cases to a contradiction if
appUed without restriction; we call this contradiction the antinomy of the
name-relation (§ 31). It is not difficult to eliminate the contradiction; various
ways have been proposed by logicians, but all of them have certain drawbacks.
The method of Frege is discussed in detail (§§ 28-30). Its main feature is the
distinction between the nominatum and the stnse of an expression. In many
cases these are the same as what we call the extension and the intension, re-
spectively. However, in contradistinction to these latter concepts, the nomina-
tum and the sense of an expression vary with the context in which the expres-
sion occurs. It is found that Frege’s method, if applied consistently, leads to an
infinity of new entities and new expressions as names for them and thus results
in a very complicated structure of the object-language. This holds still more for
the variant of Frege’s method proposed by Cburch. Russell and Quine avoid
the antinomy by not regarding as names certain expressions (although these
expressions are, in our method, L-equivalent to other expressions, which' they
do regard as names) ; thus they require an unnecessary restriction of the field
of application of semantical meaning analysis (§ 32). The fact that all forms of
the method of the name-relation lead to complications or restrictions makes it
appear doubtful whether this method is a suitable method of semantical
analysis.

§ 24. The Name-Relation

The customary method of meaning analysis regards an expression as a
name for a (concrete or abstract) entity, which we call its nominatum. The j
method, as customarily used, is based on three principles, usually implicit: the ^
principles of univocality, of subject matter, and of interchangeability.

In chapter i the concepts of equivalence and L-equivalence were intro- ^
duced and discussed, together with the derivative concepts of the extent-
sion and the intension of an expression. These concepts have been pro--
posed as tools for a semantical analysis of meaning. With our method of

96

4 24. THE NAME-RELATION

97

extenaon and inte^cm we shall now contrast that method of analysis
which seems to be accepted by many, probably by most, logicians; it is
characterised by using as basic concept the name-relation. In the present
chapter the assumptions imderlying this method of the name-relation will
be made explicit, and the consequences of its use investigated- It will be
shown that the method leads to certain difficulties, one of which will be
called the antinomy of the name-relation. Some of these difficulties have
been recognized by several logicians, and various ways have been pro-
posed to avoid them, thus leading to different forms of the method of the
name-relation. An examination of these forms will show that each of them
has serious disadvantages, e.g., an intrinsic ambiguity in the terms used,
an unnecessary multiplication of the entities leading to a complicated .
language structure, or unnecessary restrictions in the construction of lan-
guages. It will be seen that the method of extension and intenaon is free
of the shortcomings which the customary method of the name-relation
shows, at least in its known forms.

The name-relation is customarily conceived as holding between an ex-
pression in a language and a concrete or abstract entity (object), of which
that expression is a name. Thus this relation is, in our terminology, a .
semantical relation. Various phrases are used to express this relation, e.g.,^
‘x is a name for y’, ‘x denotes* y', ‘x designates y’, ‘x is a designation for
y\ ‘x signifies y’, etc. In this book I shall sometimes also use, besides ‘x
is a name of y’, ‘x names y’; this shortened form will not lead to any
ambiguity, since its customary meaning (‘ a person names an entity’) will
hardly occur here. It is often convenient to have a short term for the con-
verse relation; I shall often say, instead of ‘the entity named by (the ex-
pression) x’, ‘the nominatum of x’; I shall use this term also in formu-
lating the conceptions of other authors who do not use it.

% Logicians seem to differ widely with respect to the question of the
kinds of expressions which may be regarded as names. Nearly all will

' 3 -ase denotes y' is often used in a quite different sense, namely, in the case where x
tor for a certain property (e.g., the word ‘human’) and y is an entity having that
«PPe :y (e.g., the man Walter Scott). This semantical relation is of a rather special kind,
since it is applicable not to designators in general but only to predicators and, moreover, only
iRo predicators of degree one, unless one is willing to regard a sequence of entities as the entity
Proenoted. As a term for this relation, perhaps 'x applies to y’ and the corresponding noun ‘ap-
plication’ might also be considered. In any case, the word ‘denotes’ is at present used by many
logicians in the sense of the name-relation (see Church, [Dictionary], p. 76). Russell ([Denot-
ing]) has used the word in this sense both for the formulation of his own conception (he uses,
for instance, the term ‘denoting phrases’ for descriptions and similar expressions) and as a
translation for Frege’s term ‘bezeichnet’ (see below, { 28, n, zz). Church likewise uses this
word for the formulation of his conception, which is based on Frege’s. Following Russell and
Church, I used the word ‘denotes’ for the name-relation in the first version of this book. How-
ever, in view of the ambiguity just described, I cow prefer to avoid it.

98

m. THE METHOD OF THE NAME-RELATION

include words like ‘Napoleon’ or ‘Chicago’; perhaps a majority also words
like ‘green’ (or ‘greenness’), ‘hou^’, and ‘seven’; many also (declarative)
sentences. Let us disregard at present these differences in the domain of
application of the relation and look, rather, at the way in which it is ap-
plied. It seems to me that many logicians use the name-relation for-
semantical discussions, that is, for speaking about expressions and their
meanings, in such a way that the following three principles are fulfilled.
If an author fulfils these conditions, then we shall say that he uses the
method of the name-relation, irrespective of the terms he may use
for the relation. Sometimes an author may state* the principles ex-
plicitly; more often we shall have to infer from the use he makes of^he
relation that he regards these principles as valid. «

The Principles of the Name-Relation

24-1. The principle of univocalUy. Every expression used as a pame (in a'
certain context) is a name of exactly one entity; we call it the nominatum
of the expression.

24-2. The principle of subject matter. A sentence is about (deals with, in-
cludes in its subject matter) the nominata of the names occurring in it.
24-3. The principle of interchangeability {or
This principle occurs in either of two forms:

a. If two expressions name the same entity, then a true sentence re-
mains true when the one is replaced in it by the other; in our termi-
nology (ii-ib); the two expressions are interchangeable (every-
where).

b. If an identity sentence ‘. . . = ’ (or ‘. . . is identical with ’

or ‘ ... is the same as ’) is true, then the two argument expres-
sions ‘. . .’ and ‘ ’ are interchangeable (everywhere).

The principle of univocality is, of course, applied only to a wel
constructed language without ambiguities; its fdlfilment may, indeed, be
regarded as defining univpcal.'ty in the sense of nonambiguity, ifebn-
guage of this kind may, for instance, be an artificially constructeosy^
or a modified English, where the -ordinary ambiguities al^^elimtnatOT,^
either by assigning to an ambiguous word only one of its usual meanings^
or by replacing it with several terms for the several me^ings, e.g.y‘^ 9 l 5 -
ability,’, ‘probabilityj’.) The principle of subject lAatter is rather v^gue
but sufficiently clear for our purposes. It is sometimes used for making
the third principle plausible. And, indeed, if somebody accepts the first
two principles, he will hardly reject the third. For, if ?(, and a*; have the
same nominatu^i^d if the sentence . Jf, . . s.'>)s somelWng true

§ 24. THE NAME-RELATION

99

about this nominatum, then the sentence . . , saying the same

about the same nominatum, must also be true. The form b of the third
principle seems at first glance not to involve the name-relation at all.
But it does so implicitly in the concept of identity sign or identity sen-
tence. The following definitions of these concepts, it seems to me, are
tacitly presupposed in 24-3b:

24-4. Definitions.

a. A predicator SI, is an identity expression (for a certain type) =Df
for any closed expressions (names) SI,- and Sljt of the type in ques-
tion, the full sentence of SI, with SI, and 91* as argument expres-
sions (i.e., S[,(9Ij, 3Ii) or (9Ij)9I,(9It)) is true if and only if Slyand
SI* name the same entity.

b. is an identity sentence =Df ®t is a full sentence of an identity
expression.

On the basis of these definitions, form b of the principle of interchange- •
ability follows immediately from form a. Thus, granted the adequacy of
these definitions, form b is just as plausible as form a. I think that
Church’ expresses the generally accepted conception when he says that
the interchangeability of synonymous expressions, i.e., those which
name the same entity, follows from “what seem to be the inevitable
semantical and syntactical rules for ‘ = ’ ”.

We find an example of the method of the name-relation in Frege’s pro-
cedure. His distinction between nominatum and sense will later be dis-
cussed in detail (§§ 28-30) .'^e formulates the principle of interchange-
ability in the first form (24-3a) in this way:^

24-6. “The truth-value of a sentence remains unchanged if we replace
an expression in it by one which names the same [entity].’’

Another example of this method is Quine’s analysis in [Notes]; he uses
the terms ‘designates’ and ‘designatum’ in the sense of our ‘names’ and
^‘nominatum’. "^e principle of interchangeability in the second form
4p4-j^) is called by him the principle of substitutivity and is formulated
in this way:

24-6. “Given a true statement of identity, one of its two terms may be
substatuted for the other in any true statement and the result will be
true.’’^

This principle is not meant by Quine as a conventional rule for an identity
» sign in an artificial system but rather as an explicit form ulation of a pro-

} [Sinn], p. 36. < ‘ ‘3-

” [Keview '

D 300.

100

III. THE M^FBUM) of the NAME-RELATION .

^Itedure which is customarily" applied in the ordinary w^ l 8 %t(ap l&e
basis of the customary interpretation of the wordsT^ine distingui^es
between the designatum of an expression and its meaning; this distinction
is, as Church’ has seen, in some respects very similar to Frege’s.

The differences between the method of the name-relation and the
method of extension and intension will later be discussed in detail. Here I
wish to make only a few remarks in connection with the three principles.
The concept of the extension of an expression is, as we shall see later, in
some respects similar to the concept of its nominatum. Therefore, let us
see to what extent analogues of the three principles hold for the concept
of extension. The analogue of the principle of univocality holds; every
designator has exactly one extension. The analogue of the principle of sub-
ject matter holds, too, but with restrictions. In general, a sentence con-
taining a designator H, may be interpreted as speaking about the exten-
sion of SIj. However, it may be interpreted alternatively as speaking about
the intension of and, as we shall s^ later, the latter interpretation is
sometimes more appropriate. The decisive difference emerges with respect
to the principle of interchangeability. For extensions, instead of the ana-
logue of 24-3a, only the restricted principle 12-1 holds. It says that, if two
expressions have the same extension, in other words, if they are equiva-
lent, then they are interchangeable in extensional contexts. The principle
24-3b speaks about identity. However, on the basis of the method of ex-
tension and intension, we cannot simply speak of identity but must dis-
tinguish between identity of extension and identity of intension, in
other words, between equivalence and L-equivalence. Therefore, instead
of the one principle 24-3b for identity, we have in our method two
principles, one for equivalence and the other for L-equivalence; these are
fS-i and 12-2.

§ 26. As Ambiguity is the Method of the Name-Relatios

A predicator in a word language (e.g., ‘gross’ in German) or in a symbolic
language (e.g., an abstraction expression in Quine’s system) may be regarded
the name of a class but also as the name of a property. This shows an intrinsic
ambiguity in the name-relation. Its consequences will be discussed later.

I shall now examine in more detail some features of the method of the
name-relation, and especially try to show that the basic concept of this
method is not so simple, clear, and unambiguous as it is usually supposed
to be. ^

It seems generj^^to be assumed that, if we understand an expression,

* (Review Q.l, p. ,

525. AN AMBIGUITY IN THE NAME-RELATION loi

we know at least to what kind of entities its nominatum belongs, and
also in some cases which entity is the nominatum, although in other cases
factual knowledge is required for this. For instance, if we understand
German, then we know that the word ‘Rom’ is a name of the thing Rome,
and that ‘drei’ is a name of the number Three. In the case oF'det Autor
von Waverley’ we know at least that it names, if anything, a (physical)
thing; and if we have sufficient historical knowledge, we know that it is a
name of the man Walter Scott. Analogously, in the case of ‘die Anzahl der
Planeten’, we know at least that it names a number and, with the help of
astronomical knowledge, we know that it names the number Nine. Gen-
erally speaking, given a full understanding of the language in question
and, in particular, of some name in it and, in addition, all the factual
knowledge relevant to the case in question, we should expect that there
could be no doubt or controversy as to the nominatum of the name. How-
ever, it will now be shown that this is, in general, not the case.

Let G be a part of the German language, restricted to declarative sen-
tences, with all dubious expressions and ambiguities eliminated (see ex-
planation of 24-1) and, in particular, with the word ‘gross’ confined to its
literal meaning concerning spatial extension, Wq imagine two logicians,
L, and L„ interested in the semantical analysis of G. Before they begin
the theoretical analysis, they make certain in a practical way that they
have the same interpretation or understanding of the language G; for in-
stance, each agrees with any translation the other makes of a sentence of
G into English. Then they begin their semantical analysis of G, according
to the method of the name-relation based on the three principles (24-1,
2, 3). They examine the sentence in G; ‘Rom ist gross’. They have no
doubt and no disagreement as to its meaning; this is shown by the fact
that both agree that its translation into English is: ‘Rome is large’. Now
they apply to the expressions in the given sentence the analysis in terms
of the name-relation. Both agree that ‘ Rom’ in G is a name of the thing
Rome. But now suppose that with respect to the word ‘gross’ (or the
phrase ‘ist gross’) the following controversy arises: L, says: “The sen-
tence ‘Rom ist gross’ means that Rome belongs to the class Large. Hence
it is about the thing Rome and the class Large. Therefore, according to
the principle of subject matter, ‘gross’ is a name of the class Large; and
hence, according to the principle of univocality, it cannot be a name of
any other entity”. Against this, L, says: “The given sentence means that
Rome possesses the property Large. Hence it is about the thing Rome and
the property Large. Therefore, according to the Mjmciple of subject
matter, ‘gross’ is a name of the property Large; anoH^ce, according to

102

m. THE METHOD OF THE NAME-RELATION

the principle of univocality, its nominatum cannot be any other entity;
in particular, it cannot be the class Large.”

We might perhaps try to reconcile the two logicians by pointing out
that it does not really matter whether they say ‘ the sentence means that
Rome belongs to the class Large’ or ‘the sentence means that Rome has
the property Large’, since these two assertions are both true and differ
merely in their formulation. But, even if the two logicians were willing
to agree with us on this point, the controversy concerning the nominatum
of ‘gross’ would not be solved. Here, in distinction to the question concern-
ing the whole sentence, they cannot simply agree that they are both
right, that it does not matter whether they say that, the nominatum is the
class Large or that it is the property Large; for they agree in affirming
the principles of the name-relation; therefore they must agree, according
to the principle of univocality, that ‘gross’ (in G) can have only on^
nominatum. And, further, they agree that the class Large is not the same
as the property Large; they'agree generally in recognizing the distinction
between a property and the corresponding class, as expressed, for in-
stance, by 4-7 and 4-8. ^

Perhaps somebody will suggest to the two logicians that their insoluble
controversy is due merely to the choice of an unsuitable object language;
that a natural language like G, even after the elimination of obvious
ambiguities, is not precise enough for univocal semantical analysis; and
that, therefore, they should restrict their analysis to a well-constructed
symbolic system with exact rules. I doubt whether the controversy is
caused merely by the imperfections of G; but let us see what will result
when the two logicians follow the suggestion. Let ML be the system con-
structed by Quine in [M.L.], and ML' the system constructed out of ML
by the addition, first, of the defined signs which Quine introduces in his
book but does not count as parts of his system and, second, of a few nori-
logical atomic matrices. The two logicians agree on the following interpre-
tation of the system ML': the primitive notation of ML is interpreted in
accordance with Quine’s explanations; on this basis the interpretations of
the defined signs in ML' are determined by their definitions; for the inter-
pretation of the nonlogical atomic matrices, the following rule (similar to
1-2) is laid down:

26-1. Rules of designation (for ML').

a. ‘Hx’ is the translation of ‘x is a human thing’.^

b. ‘Fa:’— ‘a; featherless thing’.

c. ‘Ba:’ — biped thing’.

§ 25 . AN AMBIGUITY IN THE NAME-RELATION 103

‘Thing’ is here meant in the sense of ‘physical thing’. ML' is interpreted
in such a way that things are taken as individuals in Quine’s sense Ac-
cording to the rules 25-1, the three atomic matrices mentioned are fulfilled
only by entities which are things, and hence both individuals and elements
in Quine’s sense.’

The two logicians agree not to take the signs ‘H’, ‘F’, and ‘B’, intro-
duced by 25-1, as names, because it is obvious that otherwise they would
immediately get into the same controversy concerning the nominata as
they did with respect to the word ‘gross’ in G (compare the translations
4-2 and 4-3 of ‘Hs’). They agree to take as names only those expressions
which Quine calls closed terms, and among them especially the closed
abstraction expressions, i.e., expressions of the form ‘:f (. . a; . .)’ without
free variables.

Now the two logicians examine the following sentence in ML':
‘x(Ha:) C 3c(Ba:)’, which we call ©,. There is no doubt and no disagree-
ment between them as to its meaning. They agree that, according to the
rules of ML',® is L-equivalent to ‘ (x) (Hx D Bx) ’ and hence may be
translated into ‘for every x, if x is human then x is a biped’ (see 4-4; we
assume here that ‘human’ means as much as ‘human thing’, and ‘biped’
as ‘biped thing’). However, as soon as they raise the question as to what
is the nominatum of the abstraction expression ‘:C'(Hx)’, as it occurs in
®T, a controversy starts which is perfectly analogous to the earlier one
with respect to ‘ gross’ in G, in spite of the fact that we have here the exact
system ML'. L, says: “We agree about the meaning of ©i, namely,
that it is translatable as just stated; but it is likewise translatable into
‘ the class Human is a subclass of the class Biped’ (4-6). Hence ©i is about
the class Human and the class Biped. Therefore, according to the principle
of subject matter, ‘^(Hx)’ is a name of the class Human; hence, accord-
ing to the principle of univocality, it cannot be a name of any other
entity’’. L^ replies: “Since ©, is translatable" as previously stated, it is
likewise translatable into : ‘ the property Human implies (materially) the
property Biped’ (4-5). Hence ©, is about the property Human and the
property Biped. Therefore, according to the principle of subject matter,
‘:^(Hx)’ is a name of the property Human; and hence, according to the
principle of univocality, it cannot be a name of any other entity; in par-
ticular, it cannot be a name of the class Human.” Since both logicians
agree that the class Human is not the same as the property Human, they

‘[M.L.], p. 135. ’/did., p. 131.

‘ In particular, the definitions D21 and Dg in [M.L.], pp. 185 and,.^|j|||, apply here; note also
the above remark on 35-1 concerning things.

104

III. THE METHOD OF THE NAME-RELATION

must regard their statements coiiceming the nominatum of as

incompatible on the basis of the principle of univocality. In support of his
statement, L, may point to the fact that Quine, the author of the system
ML, says himself that the terms are names of classes,* that ‘ C ’ is a sign
of class inclusion,” and that the whole language deals with classes. La may
reply that he admits that the mode of speech used by Quine and by Li
can be applied consistently; his point is that the same holds for the other
mode of speech, which he uses. However, what makes the controversy
insoluble is this: The divergence between L, and Lj, which is at the start
nothing but a difference in the mode of speech, namely, between the trans-
lations of in terms of classes and in terms of properties, leads, on the
basis of the principles of the name-relation, to two statements which are
incompatible, namely, those concerning the nominatum of ‘^(Hx)’.

Now Lt discovers a new way which, he thinks, must lead to an un-
ambiguous solution of the puzzling problem. Since the difference between
classes and properties has its root in the difference of the identity condi-
tions, an identity sentence = Si,- in ML' should be analyzed where
aii and Sly are abstraction expressions; by determining the truth-condi-
tion of this sentence, we should be able to see, he thinks, whether the two
expressions Si, and SI, are names of classes or of properties. Therefore, he
proposes to examine the following sentence in ML':‘£(Hx) = ^(Fx • Bx)',
which we call There is again complete agreement between the two
logicians as to the meaning of this sentence. They agree that, according to
the rules of ML'," the sentence is L-equivalent to ‘ (x) (Hx = Fx • px) ’
and hence, on the basis of the biological fact 3-6, is true. Further, both
agree that the two classes in question are, in fact, identical (see 4-7),
while the two properties are not (see 4-8). Now L, argues as follows: “The
identity sentence ©, can only refer to the two classes; for, if it referred
to the two properties, it would be false because they are nonidentical”. L,
replies: “You, like the author of the system, take ‘ = ’ as a sign of identity
of classes. I admit that this is in accordance with the rules of the system
ML'. But then, ‘ = ’ may j^st as well be called a sign of equivalence of
properties (like ‘ = ’ in S,; see remark on 5-3). And since the two proper-
ties in question, though not identical, are indeed equivalent (see 5^5),
is also true on the basis of this analysis, which interprets the two!'Ab-
straction expressions as names of properties.”

Li will perhaps ask whether the character of ‘ ’ jn the system ML' as

’[M.L.], p. HQ.

'“Ibid., p. 185.

“ See, in particular, the definitions Dio and Dp in [M.L.], pp. 136 and 133.

§ 25 . AN AMBIGUITY IN THE NAME-RELATION 105

a genuine sign of identity and not merely a sign of equivalence, like ‘ a’
in Si, is not assured by the fact that ML' contains a principle of inter-
changeability (called principle of substitutivity of identity”). To this, L,
will give a negative answer. Interchangeability on the ba^ of ‘ s’ holds
likewise in Si (see i2-3a); thus, in this respect also, ‘ ’ in ML' is like

‘ s ’ (between predicators) in S,. It is true that general interchangeability
on the basis of ‘ s’ does not hold in some systems, for example, in Si; but
it holds in all eztensional systems (i2-3a). Thus the effect of the principle!
of interchangeability in ML' (and ML) is simply to make ML' (and ML)
an extensional language like S,; the principle prevents the introduction
into ML' of intensional predicators or connectives, for instance, of a sign
of logical necessity (like ‘N’ in Sa, see § ii. Example II). But it does not
prevent in any way the interpretation of abstraction expressions in ML'
(or ML) as names of properties.

Now let us draw the conclusion from our examination of the contro-
versy between the two logicians. Note that this controversy is not an
instance of the well-known multiplicity of interpretations, that is, of the
fact that for a given logical system (calculus) there are, in general, several
interpretations, all of them in accordance with the rules of the system.
Li and L, apply the same interpretation to their object language G, and
then likewise to the language system ML'. Even when L, says that the
sign ‘ = ’ in ML' is a sign of identity of classes while L, says that it is a
sign of equivalence of properties, this does not show a difference in inter-
pretation but merely a difference in the choice of semantical terms used
for describing one and the same interpretation ; for equivalence of proper-
ties is just the same as identity of classes (or, speaking more exactly, ‘the
properties expressed by two predicators are equivalent’ amd ‘ the corre-
sponding classes are identical’ are L-equivalent sentences in M). That Li
and Lj apply the same interpretation to ML' (as well as to G) means that
to any given sentence in ML' they attribute the same meaning or, in
other words, the same truth-condition. The decisive point is rather this:
In spite of their agreement in the interpretation, it is possible for Li
and La to maintain different conceptions as to what are the nominata of
the names occurring — conceptions which are incompatible with each
other, though each is consistent in itself. This shows, it seems to me, that
the method of the name-relation involves an intrinsic ambiguity, inas-
much as the fundamental term of this method, namely, ‘is a name of’,

» Quine, [M.L.], § 2g, *201; for the corresponding principle with respect to the word
language, see above, 24-6.

io6 III. THE METHOD OF THE NAME-RELATION

is ambiguous, although it is generally believed to be quite clear and un-
ambiguous. This is not to say that, in general, a logician uses these
terms ambiguously, but only that several logicians may use them in
different ways. For instance, L, uses the method consistently and un-
ambiguously, and so does Lj. The trouble is that, if one logician thinks
that the results which he has found on the basis of his conception must be
accepted by everybody else, he is mistaken, because it may be that the
results do not hold for another conception of the name-relation.

'We have discussed the ambiguity only with respect tc^redicators,
^here either classes or properties may be taken as nominata^nalogously,
for a designator of another kind^^ther its extension or its intension may
be taken as its nominatum. Thus there are, in fact, many more than two
ways for using the method of the name-relation. And the multiplicity of
ways is, further, considerably increased by the fact that some logicians
take some predicators as names of classes and other predicators of the
same type as names of properties (see § 26) ; and that some logicians even
take the same expression as a name of an extension in one context and in
anot^r as a name of an intension (for example, Frege, see below, §§ 28,
29^^or the present, it will suffice to point out the great multiplicity of
different ways of using a method of the name-relation, in other words, the
many different senses in which the term ‘name' or similar terms are used.
Some of these ways will be discussed later in order to show the complica-
tions which they involve.

§ 26. The Unnecessary Duplication of Names

Many systems have different names for properties and for the corresponding
classes. This is discussed with respect to examples from the system of Principia
Mathemalica. Analyzing these names by the method of extension and intension,
we find that a name for the property Human and a different name for the class
Human have not only the same extension but also the same intension. There-
fore, the duplication of names to which the method of the name-relation leads
is superfluous.

Another consequence of the customary way 'of using the method of
the name-relation will now be discussed. The principle of subject matter
(24-2) says that if a sentence contains a name of an entity, then it says
something about this entity. And the method is usually conceived in such
a way that, conversely, if a sentence is intended to be about a certain
entity, then it must contain a name of this entity. Then it follows, in
virtue of the principle of univocality (24-1), that, in order to speak about
two different entities, we have to use two different expressions as their
names.

§ 26 . THE UNNECESSARY DUPLICATION OF NAMES 107

On the basis of the method of extension and intension, on the other
hand, the situation is quite different. A designator is here regarded as
having a dose semantical relation not to one but to two entities, namely,
its extension and its intension, in such a way that a sentence containing
the designator may be construed as being about both the one and the
other entity. Thus here, if a sentence is intended to speak about an entity
which is an extension, an expression is needed whose extension is that
entity; and if we wish to speak about an entity which is an intension, an
expression is needed whose intension is that entity. Therefore, in order to
speak first about a certain intension and then about the corresponding ex-
tension, this method requires only one expression, while the method of the
name-relation would require two and hence lead to an unnecessary dupli-
cation in symbolism.

This duplication can best be made clear in the case of predicators. The
method of extension and intension needs only one predicator to speak
both about a certain property and about the corresponding class. The
method of the name-relation in its customary form, however, needs for
this purpose two different expressions, a property name and a class name.
As an example, let us take the symbolic system PM constructed by White-
head and Russell in [P.M.]; PM includes not only the primitive signs but
also the (logical) signs introduced by the definitions as given by the au-
thors. Let PM' consist of PM and, in addition, a few nonlogical predica-
tors or atomic matrices. Let PM' be interpreted in the following way: The
primitive logical signs are interpreted in accordance with the explanations
of the authors of [P.M.]; the interpretations of the defined signs are then
determined by their definitions; the nonlogical signs are interpreted by

25- 1 as a rule of designation for PM'. -

The system PM' uses different expressions as names for properties
(construed as propositional functions) and as names for classes. Take, as
examples, the following four statements concerning two pairs of expres-
sions in PM':

26- 1. ‘Hx’ is a name of the property Human.

26-2. ‘x{llxy is a name of the class Human.

26-3. ‘Fi • Bx’ is a name of the property Featherless Biped.

26-4. ‘xfFx • Bx)’ is a name of the class Featherless Bijjed.

[For the present discussion we may leave aside the fact that Russell does
not assume that there are classes as separate entities, in addition to
properties; he introduces class expressions by contextual definitions on
the basis of property expressions. The problem of this and the converse

io8 m. THE METHOD OF THE NAME-RELATION

reduction will be discussed later (§-33). For our present problem it is suf-
ficient that an author speaks in his metalanguage both of properties
(qualities, propositional functions of degree one) and of classes (distin-
guished in the customary way) ; that he uses in his object language two
different kinds of expressions; and that he declares that those of the first
kind are meant as expressions of properties and those of the second kind
as expressions of classes.]

The four statements given express results of a semantical analysis of
certain expressions in PM', according to the method of the name-relation.
If, instead, we analyze PM' by the method of extension and intension, we
arrive at the following results, which contain counterparts of the earlier
results, supplemented by new ones. Instead of 26-1, we have here;

26 -fi. The intension of ‘Hf ’ is the property Human.

To this statement, however, another statement is added, which follows
from it:

26 - 6 . The extension of ‘Hf’ is the class Human.

Instead of 26-a, we have here;

26 - 7 . The extension of ‘^(H*)’ is the class Human-
To this we add;

26 - 8 . The intension of (Hr)’ is the property Hunian. ^

t

While 26-6 follows directly from 26-5, the same is not true for' 26-8 and
26-7; every intension uniquely determines an extension, but the converse
does not hold. Statement 26-8 is based, rather, on the rule 25-ia and
the circumstance that, according to the rules of PM', the sentence
‘(y)[yei(Hr) ^ Hy]’ is L-true in PM'. The results corresponding to
26-3 and 26-4 are, of course, analogous.

Thus the outcome, from the point of view of our method, is that the
two expressions ‘Hie’ and ‘^(Hr)’ in PM' have the same extension and
also the same intension. Therefore, it is unnecessary to have both forms
in the system. The two expressions are, in a certain sense, L-equivalent
predicators. It is true that one of them cannot simply be replaced by the
other; this is the effect of certain restricting rules concerning the two
kinds of predicators. First, there is the following unessential difference,
which is merely an accidental syntactical feature of the systems PM
and PM'. The rules require that an argument expression for a predicator
of the first kind (e.g., ‘H’ or‘H£’) succeeds it (resulting in ‘Hs’), while one
for a predicator of the second kind precedes it with a copula ‘ e’ interposed
(e.g., ‘se;^(Ha;)’). Another difference is more important. It concerns

1 26. THE UNNECESSARY DUPLICATION OF NAMES 109

identity sentences built with ‘ » Consider the following two sentences as
examples;

26 - 9 . - ^(Fa; • B*)’.

26 - 10 . F£*Br.

According tii) the explanation given in [P.M.], the sentence 26-9 says that
the two classes in question are identical; hence this sentence is true (see
4-7). On the other hand, the sentence 26-10 says that the two properties
in question are identical; hence this sentence is false (see 4-8). Thus, 26-9
is in notation and meaning just like a sentence in ML' previously dis-
cussed (®, in § 25). Likewise, its L-equivalence to ‘(*)'(Ha: s Fx • Bx)’
holds for PM'. Therefore, the contention of La that ‘ = ’ in 26-9 is like ‘ = ’
in S, (or Sa) and, hence, is simply a sign of equivalence applies here as
well. On the other hand, ‘ = ’ in 26-10 is a sign of identity or L-equivalence
of properties; it is therefore, in distinction to ‘ = ’ in 26-9, a nonexten-
sional sign. (This is recognized by Whitehead and Russell.)’’ Hence it
cannot correspond to any sign in the extensional language S, ; but it cor-
responds exactly to the modal sign ‘ s ’ in Sa, which will be introduced
later (see 39-6; accordingly, the false sentence 26-10 is L-equivalent to
42-2bA without the sign of negation). Thus the method of extension and
intension by no means overlooks the difference between 26-9 and 26-10.
On the basis of this method, in distinction to the method of the name-
relation, the first components in the two sentences (i.e., the predicators
‘Hf’ and ‘i(Hx)’) are equalized in certain respects, and so are the second
components. Nevertheless, the difference is preserved because the occur-
rences of ‘ = ’ in 26-9 and in 26-10 are here construed as having different
meanings. The first is interpreted as a sign of equivalence or, in other
words, of identity of extensions; the second as a sign of L-equivalence
or, in other words, of identity of intensions.

We see that the situation with respect to the two methods under dis-
cussion is this: At the beginning, there is merely a difference of pro-
cedure in describing the semantical features of given language systems.
The customary method does it in terms of nominata; our method does it,
instead, in terms of extensions and intensions. At first glance, one might
think that both methods were neutral with respect to the structure of the
language systems, in the sense that either method is as applicable to any
system as the other. If so, the choice of the one or the other method of
semantical analysis would not have any effect upon the choice of a struc-
ture for a system to be constructed. However, this is not so. According to

IP.M.J, I, 84.

no

III. THE METHOD OF THE NAME-RELATION

the first method, the two expressions ‘H.£’ and '^(Hjc)’ are said to have
different nominata; and this circumstance is then naturally regarded as
justification for the decision to incorporate both expressions into the sys-
tem, as is done in the system PM'. According to the second method, on the
other hand, the two expressions are said to have the same extension and
the same intension. This leads to the view that the inclusion of both
would be an unnecessary duplication, and hence to the decision to con-
struct the system in such a way that it contains, instead of those two ex-
pressions, only one, as in the systems S, and Sj (and in many systems con-
structed by other logicians'^). Corresponding to the two expressions in
PM', S, and have the one predicator ‘ { Xx) (Hx) ’ (of course, either of

.the two notations in PM' could be taken, instead, just as well). That we
could do in previous examples (e.g., 3-8) without lambda-expressions was
merely due to the simplicity of the examples. In general, an identity sen-
tence for classes in PM' (like 26-9) will be translated into S, and S2 in the
form ‘(Xx)(. . .) s (Xx)( — )’, and the corresponding identity sentence
for properties (like 26-10) will be translated into Sj in the form
‘(Xx)(. . .) s (Xx)(- - -)’, with the same two lambda-expressions as the
first sentence.

Our conclusion that the duplication of predicators in PM and PM' is
unnecessary holds likewise for systems which use two different kinds of
operators for class abstraction (e.g., ‘i(. . x . .)') and for functional ab-
straction,'^ that is, formation of abstraction expressions for properties,
here construed as propositional functions (e.g., ‘(Xx)(. . x . .)’). Here
again, if the same matrix ‘ . . x . .’ occurs as scope in both expressions, they
have the same extension and the same intension ; however, they have dif-
ferent conditions of identity. Thus they are analogous to ‘:£(Hx)’ and
respectively, in PM'.

^<^'^ince the choice of a semantical method and the choice of a form of
language are interconnected, we may also reason in the inverse direction :
our preference for a language structure may influence our preference for
one of the two semantical methodsyTf a language system with only one
kind of predicator is, in fact, not only as effective (for the purposes of both
mathematics and empirical science) as a system with two kinds like PM'

That it is unnecessary to have special class expressions in addition either to simple
predicator signs and their combinations or to property expressions has already been seen by
several logicians. Concerning the historical development of this insight and concerning the
possibility of a form of language without special class expressions, see [Syntax], §§ 38 and 37.
The discussion in the present book confirms this conception by basi ng i t on a more general con-
ception, namely, that of the method of extension and intension for designators in general

's See, for instance, Church, [Dictionary], p. 3.

§ 27. NAMES OF CLASSES

III

but also simpler and hence more convenient, then I think the method of
the name-relation must be regarded as at least misleading, if not in-
adequate.

§ 27. Names of Classes

A name for a class must be introduced by a rule which refers to exactly one •
property; otherwise, the meaning of the new sign and of the sentences in which
it occurs is not uniquely determined. This shows that a semantical rule for
a sign determines primarily its intension; only secondarily, with the help of
relevant facts, its extension. The customary use of difierent kinds of variables
for properties and for classes is shown to be as unnecessary as that of difierent
names. The duplication of names and variables on the first level leads to a still
greater multiplication of names and variables on higher levels. The concepts of
mathematics can be defined without the use of special class expressions and
class variables. This is shown by definitions of ‘ 2 ’ and of ‘cardinal number’.

We have seen in the preceding section that those expressions in the sys-
tem PM' which are regarded as names of certain classes by the authors of
the system do not only have these classes as their extensions but, at the
same time, have certain properties as intensions (see 26-8)‘i^ere the ques-
tion might be raised as to whether it could not happen in some system that
a predicator has only an extension, not an intension ; in other words, that
it refers to a class without referring to any of those properties which have
that class as an extensioiJr^think that this is not possible in a semantical
system, that is, in a system whose interpretation is completely given. To
begin with, it is not possible to refer to a class without referring to at least
one of the corresponding properties. This holds, even if the class is speci-
fied by an enumeration of its members, e.g., by a phrase like ‘the class of
the individuals a, b, and c’, or in the symbolic language Si: ‘ (Xa:)[(a: = a)

V (a; = b) V (a: = c)]’. This predicator does not lack an intension; it
is the property of being (identical with) either a or b or c. The feeling
which we might have, that this is not a property in the same sense as
properties like Blue or Human, is right; it is (if ‘a’, ‘b’, and ‘c’ are inter-
preted as L-determinate constants for positions in an ordered domain,

§ 19) a positional, not a qualitative, property; in our earlier terminology
(§ 22), it is an L-determinate property; but, in any case, it is an intension.

One might perhaps think a class name without an intension could be
introduced into a system by stipulating that it is to be a name for the class
which such and such equivalent properties have in common; this reference
to several properties would have the effect that none of them would be
the intension of the name. Consider, for instance, the following as a
semantical rule for the class name ‘K’ in Sj:

IIS

III. THE METHOD OF THE NAME-RELATION

27 - 1 . ‘K’ is to be a name at once for the class Human and for the class
Featherless Biped, which is the same class.

This rule does not involve an inconsistency, since the classes mentioned
are indeed identical (see 4-7). However, it is not sufficient as a semantical
rule for ‘K’; the interpretation of ‘K’ or, in ordinary words, its meaning,
is not completely given by 27-1 but merely confined to certain possibilities.
It is true that this rule, together with rules for the other signs in S, and
knowledge of the relevant facts, is sufficient to determine the truth-value
of any sentence in S, in which ‘ K’ occurs. For instance, ‘ Ks’ is found to be
true in S, on the basis of the historical facts which make the two sentences
‘Hs’ and ‘Fs • Bs’ true. The decisive point is that, although the truth-
values, the extensions, of the sentences containing ‘ K’ are determined,
their intensions are, in general, not. For instance, it remains undetermined
what proposition is expressed by ‘Ks’; is it the same as that expressed by
‘Hs’, or by ‘Fs • Bs’, or by their disjunction, or their conjunction? These
are four different propositions. To express it in othe^erms, the given
K-rule (27-1), together with the rules for other signsydoes not suffice for
the application of the^-concepts to the sentences containing ‘K’. For in-
stance, it is not determined whether ‘Ks s Hs’ is L-true or F-true. There-
fore, strictly speaking, on the basis of the K-rule and the other rules we
cannot understand sentences like ‘Ks’ or ‘Ks = Hs’, although we can
establish their truth-values.M^e reason for the objection here raised
against the K-rule is not the fact that it introduces ‘ K’ as a name for a
class, but rather the fact that it does not do this by reference to exactly
one property. In contradistinction to 27-1, the following would be a com-
plete semantical rule for ‘K’:

27 - 2 . ‘K’ is to be a name for the class Human.

For this would say the same as: ‘ ‘K’ is to be a name for the class which
is the extension of the property Human’; and this, in turn, may be under-
stood as saying: ‘ ‘ K’ is to be a sign whose intension is the property Hu-
man; therefore, its extension is the class Human.’ The first part of this
last sentence would suffice as a rule; the second part (‘therefore . . .’) is
a semantical statement following from the rule.'’'^is shows that the
semantical rule for a sign has to state primarily its intension; the extension is
secondary, in the sense that it can be found if the intension and the rele-
vant facts are given. On the other hand, if merely the extension were
given, together with all relevant facts, the intension would not be unique-
ly determined.

§ 27. NAMES OP CLASSES

113

We have seen in the preceding section h*w the method of the name-
relation leads to the use of two kinds of predicators within the same type
(for example, level one and degree one). On the basis of this method, es-
pecially of the principle of subject matter, this duplication of predicators
is regarded as necessary if we wish to speak both about classes and about
properties. An analogous situation arises with respect to variables. For
speaking about particular entities, names are used ; and thus the method
leads t« class names and property names. On the •ther hand, f«r speaking
about entities of some kind in a general way, variables are used; thus here
the method of the name-relation leads to the introduction of two kinds of
predicator variables for the same type; the values of variables of the first
kind are classes, the values of those of the second are properties. Thus,
for example, the system PM uses ‘a’, ‘/S’, etc., as class variables and
‘0’, etc., as variables for properties (propositional functions). From
the point of view of the method of extension and intension, this duplica-
tion is analogous to that of closed predicators and just as superfluous. In
the system PMV^(Hx)’ is a value expression for ‘a’. We have seen that,
on the basis of our method, ‘ :?(Hx)’ has not only an extension, namely, the
class Human (see 26-7), but also an intension, the property Human (see
26-8). Therefore, not only does the class Human belong to the value ex-
tensions of ‘a’ according to lo-i, but it is also the case that the property
Human belongs to the value intensions of ‘a’ according to 10-2. But
exactly the same holds for ‘0’ because of 26-6 and 26-5, since ‘H^’ is a
value expression for ‘ 0 ’. Thus both kinds of variables have the same value
extensions, namely, classes of individuals, and the same value intensions,
namely, properties of individuals. Therefore, the duplication of variables
is as unnecessary as that of closed predicators. It is sufiicient to use one
kind of variable for the predicator type in question ; their value extensions
are classes, their value intensions are properties' (see § 10). Therefore, they
serve for speaking in a general way both about classes and about proper-
ties. [Thus, for instance, with respect to the examples in .§ 10 preced-
ing lo-i, sentences of both the forms (ii) and (iii) are translated into
a symbolic language with the help of the same variable ‘f in the form

‘(a/)(.

The situation with respect to variables of other kinds is theoretically
the same but practically different; while many logicians use different vari-
ables for classes and for properties, it seems that hardly anybody pro-
poses to use different variables for propositions and for truth-values, or

114

III. THE METHOD OF THE NAME-RELATION

different variables for individuals and for individual concepts. Thus our
method does not deviate here from the customary procedure.

If the reasoning on the basis of the method of the name-relation, which
leads to the use of two kinds of predicators within the simplest type, is
carried to higher levels, then it results in an immense multiplication of
predicators of the same type. From our point of view this multiplication
is as unnecessary as the duplication with which it starts. For the sake of
simplicity, let us restrict the discussion to predirAtor.'i of deg ree one, that
is to say, let usspeak only of classes and DrODe rti£S._ka ying relations
asic le. If on the ^irst level a distinction is made between names of classes"
anci names of properties, then, on the second level, four kinds of predica-
tors must be distinguished, namely:

names of classes of classes
names of properties of classes
names of classes of properties
names of properties of properties

To form examples in the system PM, let us start with the following
matrix, which contains the class variable ‘a’ as the only free variable:

‘(3a:)(3y)[~ (a; = y) • ( 2) (2 e a. = : 2 = x. V .2 = y)]’.

As shorthand for this in the subsequent examples, let us simply write
‘ . a . This matrix says that the class a has exactly two members, or,
as we may say for short, that a is a pair-class. Let ‘ . . 0 . .’ be taken as
shorthand for that matrix in PM which is analogous to the one mentioned
but which contains the property variable ‘ 0’ instead of ‘ a’ (that is to say,
‘zea’ is replaced by ‘ 1 ^ 2 ’). Hence, says that there are exactly

two individuals which have the property <^, or, as we may say, that 0 is
a pair-property. Now let us examine the following four expressions in
PM:

(i) ‘a(..a..)’,

(ii) a . .’,

(iii) ‘0(. . 0 ,

(iv) . 0 .

where the dots indicate the matrices just described. Expression (i) is a
name of the class Pair-Class and hence belongs to the first of the four kinds
of predicators on the second level mentioned above; (ii) is a name of the
property Pair-Class and hence belongs to the second kind; (iii) is a name

§ 27 . NAMES OF CLASSES

115

of the class Pair-Property and hence belongs to the third kind; (iv) is a
name of the property Pair-Property and hence belongs to the fourth kind.
The nominatum of (i) , that is, the class of all classes which have exactly
two members, is in PM taken as the cardinal number Two, and therefore
‘ 2’ is introduced as abbreviation for (i). The expressions (ii), (iii), and (iv)
do not, it seems, actually occur in the book [P.M.], but they are formed
according to the rules of the system PM. The four expressions belong to
the same type; they are predicators of level two and degree one. If we were
to construct, on the basis of our method of extension and intension, a
system with a predicator variable then it would contain, instead of the
four expressions of PM, only one, namely, ‘(X/) (. . / . .)’.

The multiplication of kinds of predicators on the basis of the method
of the name-relation increases with higher levels. On the level «, there are
2” different kinds of predicators within the same type. They are supposed
to be required as names of 2" kinds of entities. On the basis of our method,
there is only one kind of predicator in each type; and the 2" corresponding
predicators in the other method are here replaced by one.

On the basis of our method, all the mathematical concepts can be
defined in a way that is analogous to that in [P.M.] except that no special
class expressions and class variables are used. Let us suppose that 5 is a
system which contains not only individual variables but also variables for
which predicators of various levels can be substituted, say ‘/’ and ‘g’ as
variables of level one and ‘m' and ‘n’ as variables of level two. Then, for
example, the cardinal number Two can be defined in 5 as a property of
properties as follows :

27 - 3 , ‘2’ for ‘(X/)[( 32 :)( 3 y)[~(x = y) • iz)(fz = {z = x)M {z s y))]]\

It is true that a certain requirement of extensionality must be f ulfill ed
by any explicatum for the concept of cardinal number in order to be ade-
quate. However, it is not necessary to require that the cardinal numbers
be extensions; it is sufficient to require that any statement attributing a
cardinal number to a given property (or class) be extensional. This re-
quirement is also fulfilled by our method, because the cardinal numbers
are here defined as properties of properties which are extensional. That, for
example, 2 as defined by 27-3 is an extensional property of properties is
not explicitly stated in the definition, but it is seen from the fact that the
following sentence is provable with the help of the definition 27-3:

'(/)(«)[(/- ^) 3 (2^) - 2(g))]’-’

1x6

m. THE METHOD OF THE NAME-RELATION

The general concept of cardinal number can likewise be defined in the
system S without the use of special class expressions. While Russell
explicates cardinal numbers as classes of classes, Frege takes them as
classes of properties. Since we wish to take them as properties of proper-
ties, we may follow Frege’s procedure half the way. We say, like Frege,**
that the property / is equinumerous to the property g (in s)Tnbols:
‘Equ(/, g)’) if there is a one-to-one correlation between those individuals
which have the property / and those which have the property g. Then we
define the cardinal number of the property / as the property (of second
level) Equinumerous To f:

27-4. ‘Ncr foT‘{\g)[Eqn(g,f}Y.

[Frege takes as definiens not ‘the property Equinumerous To but ‘the
extension of the property Equinumerous To f, which means the same as
‘the class Equinumerous To/’. Now it is interesting to see that Frege
adds to this definition a footnote {op. ciL, p. 8o) which says: “I believe
that instead of ‘ extension of the property’ we might say simply ‘ property’.
But two objections would be raised: ... . I am of the opinion that both
these objections could be removed; but that might lead here too far.”
Thus Frege considers here the simpler procedure which we now adopt. He
seems to regard it as feasible but does not pursue it any further. In his
later work*’' he again defines cardinal number in the way stated above,
without even mentioning an alternative possibility. His chief reason for
regarding cardinal numbers as classes of properties rather than as proper-
ties of properties seems to be his view’* that cardinal numbers are inde-
pendent entities, in combination with his general conception that classes
are independent entities, while properties are not. However, I find his
reasoning on this question not quite clear and far from convincing.]
Finally, we define, like Frege,” ‘« is a cardinal number’ (in symbols:
‘NC(«)’) by ‘there is a property /such that n is the cardinal number of/’:

27-5. ‘NC’ for ‘(X«)[(a/)(» s Nc‘/)]’.

Suppose that the properties / and g are equinumerous. Frege shows on
the basis of his definitions that in this case the cardinal number of / is
equal to that of g. The latter statement is interpreted by him as saying
that the class Equinumerous To / is the same as the class Equinumerous
To g. Thus he explicates equality of numbers as identity. Here our defini-

“ [Grundlagen], pp. 73-79, 83^5. '* [Grundlagen], pp. 67-72.

[Grundgesetze], I, 57. p. 85.

i27. NAMES OF CLASSES

117

tion 27-4 may seem to involve a difficulty, because, even if / and g are
equinumerous, the property Equinumerous To / need not be the same as
the property Equinumerous To g. However, although these two proper-
ties, which in our method are regarded as cardinal numbers, are not
identical, they are equivalent (in the sense of 5-3; see the example 5-5).
Thus the difficulty disappears if we explicate equality of numbers as
equivalence rather than as identity and hence symbolize it by ‘ s’. Thus,
for example, the sentence

‘ the number of planets = 9’

would be translated into the system 5 as follows, if we take ‘ P’ as predica-
tor for the property Planet :

27 - 6 . ‘Nc‘P s 9’.

(The definition of ‘9’ is, of course, analogous to that of ‘ 2’ in 27-3.)

_ We have aaiH that wfi explicate card inal numbers as prope rties of
second level, in contrast to Frege and take them as class^ or

• ^cond te^l. But this formulation is a concession to the customary view
based on the name-relation, according to which a predicator is a name
either of a class or of a property and cannot refer to both of them at once.
According to the method of extension and intension, it would be more ade-
quate to say that we introduce cardinal number expressions as predica-
tors of second level and that these predicators have as intensions proper-
ties of second level and as extensions classes of second level. Thus, for
example, ‘2’ is a predicator of second level; its intension is the property
(of second level) Two, which we might call the number intension Two
or the number concept Two; and its extension is the class (of second level)
Two, which we might call the number extension Two. Since the sentence
‘Nc‘P = 9’ is true but not L-true, the predicators ‘Nc‘P’ and ‘9’ are
equivalent but not L-equivalent. Therefore, the number extension The
Number Of Planets is the same as the number extension Nine, while the
number intension The Number Of Planets is not the same as, but equiva-
lent to, the number intension Nine. Thus we see that in our method, too,
as in those of Frege and Russell, equality of numbers can be regarded as
identity of certain entities, not of number intensions but of number
extensions.

In this way the whole system of mathematics constructed on the basis
of logic by Frege and Russell can be reconstructed in a simpler form with-
out the use of class expressions distinct from property expressions and of
class variables distinct from property variables.

in. THE METHOD OF THE NAME-RELATION

■ § 28. Frege’s Distinction between Nominatum and Sense

Frege distinguishes for any name between its nominatum, i.e., the object
named, and its sense, i.e., the way in which the object is given by it. We see
from Frege’s discussion that his concept of nominatum fulfils the principles of
the name-relation stated earlier (§ 24); thus his method of semantical analysis
is a particular form of what we call the method of the name-relation. According
to Frege, the nominatum of an isolated sentence is its truth-value, and its sense
is the proposition expressed by it. However, if the sentence stands in an oblique
(i.e., nonextensional) context, then its nominatum is that same proposition.

Frege” has made a very interesting distinction between the nomina-
tum of an expression and its sense.*’ This distinction will now be ex-
plained and then, in the next section, compared with our distinction be-
tween extension and intension.*^t will be seen that in some respects there
is a close similarity between the two kinds of distinctions; and it was,
MiK^jpd, Trege’s pair of concepts that first suggested to me the concepts of
exteifeion and intension as applied to designators in general. Oi^he other
hand, we shall find differences between the two conception^^lJased chiefly
upon the fact that Frege’s conception is a particular form of what I have
previously called the method of the name- relation.

The purpose of Frege’s paper, described here in modern terminology, is
to carry out a semantical analysis of certain kinds of expressions in the
ordinary word language and to propose,^examine, and apply semantical
concepts as instruments for this analysis. His discussions seem to me of
great importance for the method of logical analysis; but, like his other
works, this paper has not found the attention it deserves. Except for
Russell, [Denoting], who has discussed Frege’s analysis in detail but re-
jected most of it, Frege’s paper seems to have been neglected for about
half a century, until Alonzo Church” began, several years ago, to point

'"[SinnJ.

” I list here the English terms which I shall use as translations of Frege’s terms, following,
in most cases, Russell, (Denoting], and Church (see n. 22). ‘Ausdrticken’ is translated into 'to
express’ (‘to connote’ might perhaps also be taken into consideration, in analogy to ‘to denote’,
although it often has in ordinary usage a quite different sense which concerns not the designa-
tive meaning component but other ones, especially the associative and emotive); ‘Sinn’ —
‘sense’ (so Church; Russell uses ‘meaning’; ‘connotatum’ or ‘connotation’ might also be con-
sidered); ‘bezeichnen’ — ‘to be a name of’ or ‘to name’ (Russell and Church: ‘to denote’; see
the remark on the ambiguity of this term in n. i, § 24); ‘Bedeutung’ — ‘nominatum’ (Russell
and Church: ‘denotation’); ‘Begriff’ — ‘property’ (Frege uses ‘Begriff’ for attributes of degree
one only; for attributes in general he uses the phrase ‘Begriff oder Beziehung’); ‘Gedanke’ —
‘proposition’ (see Church’s justification for this translation, [Review Q.], p, 47); ‘gewfihnlich
(Rede, Bedeutung, Sinn)’ — ‘ordinary’; ‘ungerade (Rede, Bedeutung, Sinn)’ — ‘oblique’;
‘Gegenstand’ — ‘object’; ‘Wertverlauf’ — ‘value distribution’; ‘Behauptungssatz’ — ‘(declara-
tive) sentence’.

“ In reviews in the Journal of Symbolic Logic, V (1940), 162, 163; VII (1942), lot ; see also
an abstract of a paper of his, ibid., VII, 47; further, more in detail, in [Dictionary] article,
“Descriptions”, [Review C.], and [Review Q.].

§ 28. FREGE’S NOMINATUM AND SENSE

119

out repeatedly the importance of Frege’s conception, defending its basic
idea while beginning to develop further the details of its application.

Frege’s distinction between nominatum and sense is made in the fol-
lowing way: Certain expressions are names of objects (this term is to be
understood in a wide sense, including abstract, as weU as concrete, ob-
jects) and are said to name (‘ bezeichnen’) the objects. From the nomina-
lum of an expression, that is, the object named by it, we must distinguish
its sense; thi ^ '> in which the nominatum is given hv the p.xnres-

sion. This is illustrated by the following example:

28-1. The two expressions ‘ the morning star’ and ‘ the evening star’ have
the same nominatum.

This holds because both are names of the same thing, a certain planet; in
other words, the following is a true statement of an astronomical fac^^:

28-2. The morning star is the same as the evening star.

On the other hand, the following holds:

28-3. The expressions ‘the morning star’ and ‘the evening star’ do not
have the same sense.

The reason for this is that the two expressions refer to their common
nominatum, that planet, in different ways. If we understand the language,
then we can grasp the sense of the expressions; for instance, we are then
aware that the sense of ‘the morning star’ is the same as that of the
phrase ‘ the body which sometimes appears in the morning before sunrise
in the eastern sky as a brightly shining point’. The nominatum is not,
however, given by the sense but only, as Frege puts it, illuminated from
one side (“einseitig beleuchtet”). To find the result 28 - 1 , more is, required
than merely to understand the sense of the expressions (namely, observa-
tion of facts).

After having explained the distinction in a general way, Frege proceeds
to apply it to sentences. In a (declarative) sentence we express a proposi-
tion (‘ Gedanke’) . Is the proposition expressed by a sentence its sense or
its nominatum? By a long and careful analysis, Frege arrives at the fol-
lowing two results:

28-4. The (ordinary) sense of a sentence is the proposition expressed by it.
28-6. The (ordinary) nominatum of a sentence is its truth value.

These are the results for ordinary cases; they hold, in particular, for any
isolated sentence, that is, one which is not a part of a larger sentence ; the
exceptions will be discussed later. For our purposes the most important

lao “ III. THE METHOD OF THE NAME-RELATION

question to be raised here concerns the method by which Frege arrives at
these two results (and at the exceptions to them). They are clearly not
meant simply as conventions, as, so to speak, part of the definitions of the
terms ‘sense’ and ‘nominatum’. If this had been Frege’s intention, he
probably would have chosen a simple general rule not complicated by
exceptions. It becomes clear from his discussion that the situation is other-
wise. Frege assumes that he knows quite clearly what he means by ‘sense’
and ‘nominatum’, that is, that he knows the way in which he intends to
use these terms. On the basis of this knowledge, he investigates how these
terms apply to various kinds of expressionsV<^rhereby he discovers objec-
tive results, and these he reports as he finds them, whether they are
simple or complicated. For the reader, however, it is not so clear as for
Frege himself what is to be understood by his two terms. The preliminary
explanations which he gives are certainly not sufficient to lead to the re-
sults, or even to make them plausible. The nominatum of an expression,
for instance, is explained as that of which the expression is a name. This
explanation, however, by no means succeeds in making the result 28-5
plausible. I think any unprepared reader would be inclined to regard a
sentence as a name of a proposition rather than as a name of a truth-
value — if, indeed, he is at all willing to regard a sentence as a name of any-
thing. Another explanation for ‘nominatum’ which Frege gives is that a
sentence is about the nominata of the expressions occurring in it (we have
previously called this the principle of subject matter, 24-2). But this
explanation, it seems to me, does not make 28-5 any more plausible. Take
as an example the false sentence ‘Hw’ (see rules i-i and 1-2) as part of
‘ '"-'Hw’. (According to Frege, this is an ordinary case, that is to say, 28-4
and 28-5 also hold for ‘Hw’ in this context.) The question here is whether
the nominatum of ‘Hw’ as part of ‘'^Hw’ is (i) falsity or (ii) the (false)
proposition that the book Waverley is a human being. According to the
principle of subject matter, the sentence ‘ '"-'Hw’ is in case (i) about falsity
(presumably saying that falsity does not hold), and in case (ii) about the
proposition mentioned (presumably saying that it does not hold). I be-
lieve that the first alternative, which is Frege’s result 28-5, would appear
to any unprepared reader far less natural than the second.

The foregoing considerations are by no means intended as refutations
of or objections to Frege’s results. They are merely meant to show that
^ rege’s preliminary explanations of his terms are not sufficient as a basis
for his results. In order to understand the specific sense in which Frege
means his terms, we have to look not so much at his preliminary explana-

§ 28. FREGE’S NOMIN^TUM AND SENSE zst

tions as at the reasoning by which he reaches his results. When we do this,
we find that Frege makes use of certain assumptions as if they were self-
evident or at least familiar and plausible, without formulating them
explicitly as the basic principles of his method. These assumptions can be
formulated as principles of interchangeability in the following way:

Frege’s Principles of Interchangeability

Let . . 21 ,- . . be a complex name containing an occurrence of the
name 2ly, and . . K* . . the corresponding expression with the name 21*
instead of 21,-.

28 - 6 . First principle. If 21 , and 21 * have the same nominatum, then
. . 21, . . and . . 21* . . have the same nominatumyih other words, the
nominatum of the whole expression is a function of the nominata of the
names occurring in it.

28 - 7 . Second principle. If 21 , and 21 * have the same sense, then . . 21 , . .
and . . 21 * . .-JlSve the same sense. In other words, the sense of the whole
expression is a function of the senses of the names occurring in it.

Now let us see how Frege reaches his results 28-4 and 28-5 with the help
of the first principle. His problem is: What is the nominatum and what is
the sense of an (isolated) sentence? He says: “If we replace a word in a
sentence by another word with the same nominatum but a different
sense, then this change canpot have any influence upon the nominatum
of the whole sentence. ”^^ere, the first principle seems to be tacitly pre-
supposed. Let us take two sentences which are alike except for the oc-
currence of the phrases ‘ the morning star’ in the one and ‘ the evening
star’ in the other. According to our earlier statements (28-1 and 28-3),
this is a case in question. Hence, according to Frege’s reasoning just
quoted, the two sentences have the same nominatum. What, then, could
be regarded as this common nominatum? The propositions expressed by
the two sentences may, obviously, be different. Hence they cannot be the
nominata; therefore, Frege reasons, they must be the senses bf the sen-
tenceS::'f?Iere another assumption seems to be tacitly made, namely, that
the proposition expressed by a sentence, because it has clearly a close
(semantical) relation to the sentence, must be either its nominatum or its
sense.) On the other hand, the two sentences have the same truth-value
(at least in ordinary cases). Therefore, the truth-value may be regarded

« [Sinn], p. 32.

132

III. THE METHOD OF THE NAME-RELATION

as the common nominatum. Thus the results 28-4 and 28-5 are reached
(for ordinary cases).

The most important application of Frege’s two principles is to cases in
which the whole expression . . ST/ . . is an isolated sentence (while 8/ may
be either a sentence or a name of another form). For these cases the prin-
ciples take the following special forms, if the results 28-4 and 28-5 are
applied to the whole sentences:

Frege’s Principles of Interchangeability within Sentences

Let . . Sly . . be an isolated sentence containing an occurrence of the
name 8y, and . . 8* . . the corresponding sentence with the name 8* in-
stead of 8y.

28 - 8 . First principle. If 8,- and 8it have the same nominatum, then
. . 8, . . and . . 8* . . have the same truth-value. In our terminology
(ii-i): Names which have the same nominatum are interchangeable with
one another.

28 - 9 . Second principle. If 8y and 8* have the same sense, then . . 8y . .
and . . 8* . . express the same proposition. In our terminology: Names
which have the same sense are L-interchangeable with one another.

Our references in what follows are to these specialized forms of Frege’s
two principles.

What Frege means by ‘nominatum’ and ‘sense’ is shown more dearly
by these principles than by his preliminary explanations. Frege’s first
principle 28-8 is the same as 24-3a, the principle of interchangeability for
the name-relation. Since Frege’s discussion shows that the principles 24-1
and 24-2 also hold for his concept of nominatum, his method is a particu-
lar form of what we have called the method of the name-relation. As we
have seen earlier, 24-3a is quite plausible; hence Frege’s first principle is
plausible. Whether this is also true for his second principle is hard to
say. But I think it does not seem implausible if we regard it as revealing
the fact that Frege understands the term ‘sense’ in such a way that the
sense of a compound expression and, in particular, of a sentence is some-
thing which is determined by the senses of the names occurring in it.

Frege’' rl' ciples lead him, on the one hand, to the results 28-4 and
28-5 for .,ry cases — for example, for isolated sentences — as we have
seen. On the other hand, these same principles compel him to regard
certain cases as exceptions to these results and thereby to make his whole
scheme rather complicated. These exceptions are the cases in which a
name occurs in an oblique context (which is about the same as a non-

§ 28. FREGE’S NOMINATUM AND SENSE i*3

extensional context in our terminology, il- 2 a). Take, for example, the
occurrence of the (false) sentence

(i) ‘ the planetary orbits are circles’
within the oblique context

(ii) ‘Copernicus asserts that the planetary orbits are circles’.

The problems involved here would, of course, be the same if, instead of
‘asserts’, a term like ‘ believes’ were to occur; hence this example is similar
to the belief-sentences discussed earlier (§ 13 ). According to Frege’s re-
sults ( 28-5 and 28 - 4 ), the ordinary nominatum of (i), that is, that nomina-
tum which this sentence has when occurring either isolated or in an ordi-
nary, nonoblique context, is its truth- value, which happens to be falsity;
and the ordinary sense of (i) is the proposition that the planetary orbits
are circles. Now Frege says that the sentence (i) within the oblique con-
text (ii) has not its ordinary nominatum but a different one, which he
calls its oblique nominatum, and not its ordinary sense but a different one,
which he calls its oblique sense. Concerning the oblique nominatum, Frege
makes the following two statements; the second is a special case following
from the first;

28-10. The oblique nominatum of a name is the same as its ordinary
sense.

28-11. The oblique nominatum of a sentence is not its truth-value but the
proposition which is its ordinary sense.

Thus, for the above example the following result holds:

28-12. The oblique nominatum of the sentence (i), that is, the entity
named by (i) in an oblique context like (ii), is the proposition that the
planetary orbits are circles.

For this result, Frege gives two reasons at different places in his paper, (r)
“In the oblique mode of speech, one speaks about the sense, for example,
of the utterance of another person. Hence it is clear that ... in this
mode of speech a word does not have its ordinary nominatum, but names
that which ordinarily is its sense.’”^ I understand Frege’s reasoning here
in the following way, if applied to the above example. He seems to presup-
pose tacitly the principle of subject matter ( 24 - 2 ). According to it, the
whole sentence (ii) speaks about the nominatum of the subsentence (i).
Now it is clear that (ii) does not speak about the sentence (i), because
Copernicus may have used other words than (i) and even another lan-
Ibid., p. 28.

124

in. THE METHOD OF THE NAME-RELATION

guage. Nor does (ii) speak about the truth-value of Copernicus’ statement
but rather about its sense, because (ii) says that Copernicus asserted a
certain sense, a certain proposition, namely, that proposition which is the
ordinary sense of (i). Therefore, this proposition must be the nomina-
tum of (i) in (ii). (2)'TTlat the nominatum of a sentence in an oblique
context is not the truth-value but the proposition, is, Frege says, “also to
be seen from [the circumstance] that it is irrelevant for the truth of the
whole sentence whether that proposition is true or false.”'® This is pre-
sumably meant in the following way: According to Frege’s first principle,
the nominatum, that is, the truth-value, of the whole sentence (ii) is a
function of the nominatum of the subsentence (i). Now if the latter
nominatum were the truth-value, then the truth-value of (ii) would depend
upon that of (iV^his, however, is not the case; in order to establish that
(ii) is true we need not know whether (i) is true or false. Hence the nomi-
natum of (i) in (ii) cannot be its truth- value; therefore, it must be the
proposition. (For this last step, again, a certain assumption seems tacitly
presupposed.)

'In one respect, Frege’s concept of proposition (‘Gedanke’) is not quite
clear; he does not state an identity condition for propositions. In the fore-
going discussion I have assumed that he takes the same identity condition
that we take, namely, L-equivalence (see § 6 and [1], p. 92). However, in
this case, Frege’s analysis of sentences with terms like ‘asserts’, ‘ believes’,
etc., is not quite correct; because a sentence of this kind may change its
truth-value and hence, a fortiori, its sense if the subsentence is replaced by
an L-equivalent one (see, for example, the discussion of belief-sentences
in § 13, especially 13-4). His analysis would be correct if he had in mind a
condition stronger than L-equivalence, something similar to the concept
of intensional structure explained above (§ 14). In this case our second
formulation of 28-9, which was meant as a translation of Frege’s second
principle into our terminology, must be omitted.

§ 29. Nominatum and Sense: Extension and Intension

Frege’s pair of concepts (nominatum and sense) is compared with our pair
(extension and intension). The two pairs coincide in ordinary (extensional)
contexts, but not in oblique (nonextensional) contexts. This does not constitute
an iueumpatibility, a theoretical difference of opinion, but merely a practical
difference of methods. Frege’s pair of concepts is intended as an explicatum for
a certain traditional distinction, and our pair as an explicatum for another
distinction.

p. 37.

§ 29 . NOMINATUM AND SENSE

i»S

We shall now compare Frege’s distinction between the nominatum and
the sense of an expression with our distinction between the extension and
the intension of an expression.

Our pair of concepts is, like Frege’s, intended to serve for the purposes
of semantical meaning analysis. Our two concepts may be regarded, like
Frege’s, as representing two components of meaning (in a wide sense).
The concepts of sense and of intension refer to meaning in a strict sense,
as that which is grasped when we understand an expression without know-
ing the facts; the concepts of nominatum and of extension refer to the ap-
plication of the expression, depending upon facts.
vA decisive difference between our method and Frege’s consists in the
fact that our concepts, in distinction to Frege’s, are independent of the
contexf^^n expression in a well-constructed language system always has
the same extension and the same intension; but in some contexts it has
its ordinary nominatum and its ordinary sense, in other contexts its
oblique nominatum and its oblique sense.

''Let us, first, compare the extension of an expression with its ordinary
nominatum; it seems that these concepts coincidd!’'^ith respect to predi-
cator^!^ rege does not seem to have explained how his concepts are to be
applied; however^HTthink that Church’^ is in accord with Frege’s inten-
tions when he regards a class as the (ordinary) nominatum of a predicator
(of degree one ) — ^ instance, a common noun — and a property as its
(ordinary) sense .*’as an example. Church states that the nominatum of
‘unicorn’ is the null class, and its sense is the property of unicorn-hood.
And here the extension is likewise the class in question. With respect to a
sentence, its truth-value is both the ordinary nominatum and the exten-
sion. And in the case of an individual expression the ordinary nominatum
and the extension is the individual in question. Thus we have this result;

> 1 ^ 9 - 1 . For any expression, its ordinary nominatum (in Frege’s method) is
the same as its extension (in our method).

It is more difficult to see clearly what constitutes the ordinary sense in
Frege’s method. As mentioned before, this is due to the lack of precise
explanation and especially of a statement as to the condition of identity
of sense; we shall assume here againVJTat Frege would agree to take
L-equivalence as this condition. Then, for a sentence, its ordinary sense is
the proposition expressed by it, hence it is the same as its intension. For
a predicator (of degree one) its ordinary sense is the property in question,
and its intension is the same. Frege does not use any special term for the

[Review C.i, p. 301.

126

m. THE METHOD OF THE NAME-RELATION

sense of an individual expression.’’. But he says that the sense of a sen-
tence is not changed if an individual expression occurring in an ordinary
context is replaced by another one with the same sense. Therefore, it
seems reasonable to assume that what he means by the sense of an indi-
vidual expression is about the same as what we mean by an individual con-
cept. Hence, on the basis of our understanding of Frege’s explanations,
the following seems to hold;

29-2. For any expression, its ordinary sense (in Frege’s method) is the
same as its intension (in our method).

Thus, for ordinary occurrences of expressions, our two concepts coincide
with those of Frege. The differences arise only with respect to expressions
in an oblique contextNjfere our concepts lead to the same entities as for
the ordinary occurrences of the same expressions, while Frege’s concepts
lead to different entities. As we have seen earlier, this complication is
“liot introduced by Frege arbitrarily but is an inevitable consequence of his
general principles, especially the first.

It seems that Frege, in introducing the distinction between nominatum
and sense, had the intention of making more precise a certain distinction
which had been made in various forms in traditional logicV<9"hus his task
was one of explication (in the sense explained in the beginning of § 2 ).
The explicata proposed by him are the concepts of nominatum and sen.se.
Now the question is: What were his explicanda, that is, for which pair of
traditional concepts did Frege propose his explicata? Church’* refers in
this connection, first, to the distinction between ‘extension’ and ‘compre-
hension’ in the Port-Royal Logic, and, second, to the disti^tion between
‘denotation’ and ‘connotation’ made by John Stuart MillTlt seems to me
that we find in the historical development two pairs of correlated concepts,
appearing in various forms. Yhese pairs are closely related to each other
antHnay sometimes even merge. N^ertheless^^think that it is, in gen-
eral, possible to distinguish themVIi) In traditional logic we often find
two correlated concepts: on the one hand, what was called the ‘ extension’
or ‘denotation’ (in the sense of J. S. Mill) of a term or a concept; on the
other hand, what was called its ' intension’, ‘ comprehension’, ‘ meaning’, or
‘connotation’.” It seems to me that Frege intended an explication of this

Church uses the term ‘description’, which is, however, more customary for an individual
expression constructed with an iota-operator than for its sense.

[Review C.], p. 301.

« For a detailed discussion and comparison of the conceptions of Mill and other authors
see Ralph M. Eaton, General Logic (igsi), chap. vi.

§ 29. NOMINATUM AND SENSE

1'27

I^Y hiR distinrtmr. hptTP»./.n. J^Kfi_i7fl1iw.-Histrihiil:inn nf

pro positional function and the propositional function itself; in the case of

pro perty.^ur distinction between extension and intension is likewise
n^ eant as an exDlicatiQ iLaf..tbf. aamg M,k,Q { concej)ts., as far as predicators
are concerned, ant j giTwnltano/MK^Iy i^e an pnlargrpmpnf ttip rlntnam mf
application of the customary con cepts to oth« kinds of designators. (2)
.The seTondTpaiFoTconcepts starts with the name-relation. In everyday
language, it is said, for instance, that ‘Walter Scott’ is a name of the man
Walter Scott. Logicians extend the application of this relation. They also
regard individual descriptions as names, e.g., ‘the author of Waverley’ as
a name of the same man Walter Scott, a usage not admitted by everyday
language. Going further, they even construe expressions of another than
the individual type as names; they regard them as names of abstract enti-
ties, e.g., of classes or properties, relations, functions, propositions, etc.
(Other terms used a^ynonyms of ‘is a name of’ were mentioned at the
beginning of § 24.) with respect to any expression regarded as a name, a
distinction is made here between that entity whose name the expression is
and the meaning or sense of the expression. It seems that the second con-
cept in this pair is very similar to the second in the first pair; for both of
them the term ‘meaning’ is sometimes used.

v^ow it seems to me that the explicandum which Frege intended to
explicate by his distinction between nominatum and sense was the second
pair of concepts rather than the first^^Cnd I interpret also some of Quine’s
discussions in [Notes] as an endeavor toward a clarification and explica-
tion of the concepts of the second pairN^iftgg^Church’s disc p a s jpns in- rtv .
cent pnhhVa firing, [R PVTPW

defend and develop Frege’s distinction, I regard them, too, as belonging
more 10 ihe decoiiTKislorical line than to the first. However, the two his-
torical lines, the two pairs of concepts takep as explicanda, are closply
related to each other. I have emphasized the difference between them only
in order to make clearer the difference between the problem which Frege
intended to solve and my problem or, more exactly, the difference between
the explicandum which Frege took as the basis of his distinction between
nominatum and sense (if I understand him correctly) and the explican-
dum for which my distinction between extension and intension is in-
tended.

^✓Thus it becomes clear — and I wish to emphasize this point-^^Eat the
difference between Frege’s method and that here proposed i§_not a dif-
ference of opinion. In other words, it is not the case that there Ts“(me

ni. THE METHOD OF THE NAME-RELATION

128

question to which different and incompatible answers have been given.
There are two questions, and, more precisely ,'^ese are not even theo-
retical questions but merely practical aims. While the general aim is the
same, namely ,*TEe construction of a pair of concepts suitable as instru-
ments for semantical analysis, the specific aims are different. Frege tries
to achieve the general aim by an explication of one pair of concepts, I by
the explication of another pair. Frege’s principles are not assertions which
are open to refutation or doubt. They are to be regarded rather as part of
the characteriza^n of his two concepts and hence'Tiold analytically for
these conceptsH^f someone were to say — as I do not — that he disagrees
with Frege’s principles, he would merely be saying in effect that he under-
stands the two terms ‘ nominatum’ and ‘ sense ’Vff a way different from
Frege — in other words, that he uses different concepts — and there would
be thus no genuine disagreement. '?tie results found by Frege, including
the complication in the case of oblique contexts, are consequences of his
principles and hence shareVheir analytic validity (assuming that Frege
made no mistake in reasoning from the principles to the results). There-
fore, I am in complete agreement with Frege’s results in this sense: they
are valid for his concepts. The same holds for Church’s results on the same
(or a somewhat modified) basis.

The two concepts used in our method coincide, as we have seen earlier,
in certain cases with Frege’s concepts, while in other cases they do not.
This is not a contradiction between two theories, since our concepts are
admittedly different from F'rege’s. The situation is, rather, similar to the
following: Suppose someone divides all animals into aquatic, aerial, and
terrestrial animals; someone else divides them into fishes, birds, and the
rest. The two classifications coincide to some extent because fishes are
aquatic animals and birds are aerial; but they do not coincide entirely.
The one man puts whales into his first class, while the other does not.
This fact, however, does not constitute a difference of opinion, a theoreti-
cal contradiction, because the two concepts in question are admittedly
different. Since the two classifications and the assertions made on their
bases are not incompatible, it would be theoretically possible to use both
simultaneously. However, if the simultaneous use of both seems unneces-
sarily complicated, there is a kind of practical incompatibility or competi-
tion. In this case the decisive question is this: which of the two triples of
concepts is more fruitful for the purpose for which both are proposed,
namely, a classification of animals?

The situation with regard to Frege’s pair of concSpts and that proposed
here seems to me to be analogous. I have the feeling, without being quite

§ 30. THE DISADVANTAGES OF FREGE’S METHOD

129

certain, that it would not be very fruitful to use simultaneously both pairs
of concepts for semantical analysis. If so, then there is, in spite of the
theoretical compatibility, a practical competition or conflict. This con-
flict might, for instance, appear over the following point, which has been
mentioned earlier ^^f'logician, thinking in terms of Frege’s concepts,
might be inclined, though not compelled, to construct a logical system in
such a way that it contains different expressions for classes and for
properties, while a logician, thinking, instead, in terms of extension and
intension, would probably be less inclined to do so.

§ 30. The Disadvantages of Frege’s Method

Frege’s special form of the method of the name-relation involves additional
complications. Starting with any ordinary name, it leads to an infinite number
of entities and an infinite number of expressions as names for them, while the
method of extension and intension needs only one expression and speaks only of
two entities. Furthermore, according to Frege’s method, the same name, when
occurring in different contexts, may have an infinite number of difierent
nominata; and sometimes even the same occurrence of a name may simul-
taneously have several nominata.

The disadvantages of Frege’s pair of concepts in comparison with the
.^ir here proposed all belong to the concept of nojjffnatum. Frege’s con-
cept of sense is very similar to that of intension ; we might even say that,
when we consider simply these two “concepts, it is difficult to see any
reason that there should be a difference between theiiir’rhe difference is
brought about by Frege’s differentiation between the ordinary and the
oblique sense of a name.'ffTs not easy to say what his reasons were for re-
garding them as different. Perhaps he was led to make this distinction be-
cause of his original distinction between the'^dinary and the oblique
nominatum. It does not appear, at least not to me, that it would be un-
natural or ijn plausible to ascribe its ordinary sense to a name in an oblique
context.MSowever, Frege could not do this because he ha^^iready used
this ordinary sense as nominatum in the oblique context/ ^nd since he._
assumes that nn minatnm a.nd sP!q,n, |Ui|bl aliiiilHH, |||, ilifffuTTlI he had thus
to introduce a third entity as the oblique sense. Incidentall vi it seems that
Trege nowhere explains in more customary ternis what this third entity is.
v^ince Frege’s method is a special form of what we have called the meth-
od of the name-relation, it also possesses the disadvantages which we
have previously found in this method. We found (§ 25) that the concept
of nominatum involves a certain ambiguity, which is also transferred to
other semantical concepts, for instance, those of identity sentence and
identity sign.

III. THE METHOD OF THE NAME-RELATION

130

Further, we saw (§§ 26, 27) that ^he method of the name-relation may
lead to a complicated duplication or multiplicity of names within the
same type. If Frege’s form of the method is adopted, the situation becomes
even more complicated. We shall illustrate this by two examples. (See the
diagram, where an arrow with ‘N’ indicates the name-relation and an
arrow with ‘S’ the sense-relation.) Example (i): Let us start with a name

Names:
Example ( i ):
Example (2);

BnHths:
Example (1):
Example (2):

ni ftj r>* .

n„ say the sentence ‘Hs’. According to Frege’s method, there is an entity,
e„ named by this name; this is the truth-value of ‘Hs’. And there is an-
other entity, e„ which is the sense of ‘Hs’; this is the proposition that
Scott is human. This proposition Cj may also have a name; if we wish to
speak abou^, we need a name for it. This name is different from n, be-
cause the latter is the name of Ci and hence, in a well-constructed lan-
guage, IS^uld not be used simultaneously as a name of another entity. Let
the new name be n,. Like any name, n, has a sense. This sense of n^ must
be different from the nominatum of nj; it is a new entity, Cj, not oc-
curring in customary analyses. In order to speak about Cj, we need a new
name, Dj. The sense of nj is a new entity 04; and so forth ad infinitum.
Example (2) ; The situation is analogous if the first name n, is of another
type, for instance, a predicator, say ‘H’. The entity e, named by n, is here
the class Human; the sense e^ is the property Human. The name n^ is
introduced as a name for the property Human; and the new entity e, is
the sense of this name. The name n^ is a name of this sense Cj ; e4 is the
sense of this name n,, and so onl'^enerally speaking, if we start with any
name of a customary form, we have, first, two entities familiar to us: its
>jf8inary nominatum and its ordinary sejjse; they are the same as its
"^tension and its intension, respectivelj^f^hen Frege’s method leads, fur-
ther, to an infinite number of entities of new and unfamiliar kinds; and,
if we wish to be able to spjeak about all of them, the language must con-
tain an infinite number of names for these entities. 'To provide for this
infinite sequence of names seems, thus, a natural decision on the basis of
Frege’s method. 'tflfiid Chixrch does, indeed, take this decision in his de-

§ 30 . THE DISADVANTAGES OF FREGE’S METHOD 131

velopment of Frege’s method by declarmg.it desirable “that the object
language should contain for every name in it a name of the associated
sense.”** On the basis of the method of extension and intension, on the
other hand, we need in the object language, instead of an infinite sequence
of expressions, only one expression (for instance, in the first example ‘Hs’,
in the second ‘H’) ; and we speak in the metalanguage only of two entities
in connection with the one expression, namely, its extension and its in-
tension (and even these are, as we shall see later, merely alternative ways
of saying the same thing) .

The fact that, according to Frege’s method, the same name may have
different nominata in different contexts has already been mentioned as a
disadvantage. But the multiplication of entities goes far beyond Frege’s
initial distinction between the ordinary and the oblique nominatum of a
name. Actually, these two nominata constitute only^e beginning of an
infinite sequence of nominata for the same namey^ we apply Frege’s
method to sentences with multiple obliqueness, then we have to distin-
guish the ordinary nominatum of the name, its first oblique nominatum,
its second oblique nominatum, and so forth. In order to construct an
example, let us suppose that the system S contains not only, like Sj (see
§ II, Example II), modal signs, say ‘Np’ for ‘it is necessary that p’ and
‘ Op’ for ‘it is possible that p’, but also psychological terms, say ‘Jp’ for
‘John believes that p’. Now let us consider a series of sentences in 5, each
occurring within the next in a simple oblique context:

(i) ‘Hs’ {‘Scott is human’);

(ii) ' 0 (Hs)’ (‘it is possible that Scott is human’) ;

(iii) ‘ J(0(Hs))’ (‘John believes thatitis possible that Scott is human’);

(iv) ‘~N(J(0(Hs)))’ (‘it is not necessary that John believes that it is
possible that Scott is human’); etc.

Let us see what the nominatum of the original sentence ‘Hs’ is in these
various contexts. According to our previous explanation of Frege’s
method, the nominatum of ‘Hs’ in isolation is its truth- value, hence the
entity e, in the above diagram; and the nominatum of its occurrence
within (ii) is the proposition that Scott is human, hence the entity Ca in
the diagram. It can further be shown, by an analysis which we shall not
describe here in detail, that the nominatum of ‘Hs’ within (iii) is Cj, its
nominatum within (iv) is 04 , and so on.'’*l1nis the same expression ‘Hs’
has an infinite number of different entities as nominata when it occurs in
different contexts.

»• [Review Q.], p. 47-

133

ni. THE METHOD OF THE NAME-RELATION

This fact — that different occurrences of a name may have different
nominata — is certainly a disadvantage. It is the reason that Church pro-
poses a certain modification of Frege’s method whereby this multiplicity
of nominata is avoided (see § 32, Method III).

Worse than the multiplicity of nominata for different occurrences of a
name is the fact tha^'wWhin certain contexts, according to Frege’s own
analysiiSy^e occurrence of a sentence has simultaneously two different
nominata. Frege takes as an example a sentence ‘ Bebel wahnt, dass . .
that is (writing ‘A’ as an abbreviation for a long subsentence), ‘Bebel has
the illusion that A’, or ‘Bebel believes erroneously that A’. Frege inter-
prets this sentence, no doubt correctly, as ‘ Bebel believes that A ; and not
A’. Now here we have two occurrences of ‘A’, the first in an oblique con-
text, the second in an ordinary one, with therefom*different nominata.
■TRius Frege comes to the collusion that, in the'Sfi^nal sentence ‘Bebel
believes erroneously that A^he subsentence ‘ A’ ‘Tmctly speaking, must
be taken twice with differqnt^minata of which the one is a proposition,
the other a truth-value”.^’ The situation is analogous in a case like ‘John
knows that A’, because this implies ‘John believes that A; and A’,
^^his double nominatum of a name, not, as in the earlier cases, for dif-
ferent occurrences but for the same occurrence, seems a startling result of
Frege’s method. The sentences in question seem perfectly clear. At first
glance it will not seem plausible that^e subsentence ‘ A’ should simul-
taneously name two distinct entities.'^can easily be seen that the feature
here discussed has nothing to do with the ordinary ambiguities so fre-
quently met with in natural word languages, ^ut is likewise to be found in
an exact, symbolic system of modal logic. A modal sign ‘CT’ for con-
tingent truth of propositions (which is a nonsemantical concept, see § 23)
can be introduced in on the basis of ‘N’ (see §11, Example II) in this
way:

30 - 1 . Abbreviation. ‘CT(/>)’ for‘/> • '^N(/>)’.

On this basis, the sentence ‘CT(Hs)’ is L-equivalent to ‘Hs • ~ N(Hs)’;
in words: ‘Scott is human, but it is not necessary that Scott is human’;
or, briefly: ‘Scott happens to be huraan’.'’lCccording to Frege’s analysis,
the sentence ‘Hs’ within ‘CT(Hs)’ has at once two different nominata, as
have the signs ‘H’ and ‘s’; and the same holds for the words ‘Scott’ and
‘human’ in the sentence ‘ Scott happens to be human’. This seems a rather
unsatisfactory result.

If, instead of Frege’s method, the method of extension and intension is

»' [Sinn), p. 48.

§31. THE ANTINOMY OF THE NAME-RELATION

*33

used, then the situation becomes much simpler. Every expression has al-
ways the same extension and the same intension, independent of the
context. The problems connected with modal contexts will be discussed
later (chap. v).

§ 31. The Antinomy of the Name-Relation

The third principle of the name-relation ( 24 - 3 ) permits replacing a name with
another name of the same entity. Although this principle seems quite plausible,
it is not always valid. This has been pointed out by Frege, Russell, and Quine.
The contradiction which sometimes arises if such a replacement is made in a
nonextensional context is called here the antinomy of the name-relation.

The principles which characterize the method of the name-relation
(24-1, 2, and 3) seem quite plau^te; and this holds for either form of the
principle of interchangeabilityTthe one using the concept of name-relation
(24-3a) and the other using the concept of identity (24-3b). Therefore, in
a naive approach without a closer investigation, we might be tempted
to regard these principles as generally valid without any restrictions. How-
ever, if we do so and, in particular, if we apply the principle of inter-
changeability in either form to nonextensional contexts, we arrive at a
contradiction. I propose to call this contradiction the antinomy of the
nam€-relation)^^y choice of this term is, of course, motivated by the fact
that, from my point of view, the method of the name-relation is respon-
sible for the antinomy .''^fliers, who regard this method as harmless and
unobjectionable and who feel that the source of the difficulty lies, rather,
in the use of modal contexts or, more generally, intensional contexts or,
still more generally, oblique (i.e., nonextensional) contexts, will perhaps
prefer to call it the antinomy of modality or of intensionality or of
obliquity.]

The antinomy of the name-relation can be constructed, as we shall see,
in either of two forms; the first uses the first form of the principle of inter-
changeability (24-3a), the second uses its second form (24-3b). The second
form of the antinomy may perhai^^^o be called antinomy of identity or
antinomy of identical nominata or antinomy of synonymity (provided the
term ‘synonymous’ is understood,^?©! in the sense of ‘intensionally iso-
morphic’ (14-1), but as ‘having the same nominatum’).

,^,,^rege was the first to point out the circumstance that the principle of
interchangeability (see 24-5) if applied to the ordinary nominata of names
does not hold for oblique contexts. Although Frege’s formulation was not
presented in terms of a contradiction, his result constitutes the basis of
what I propose to call the antinomy of the name-relation.

134

ni. THE METHOD OF THE NAME-EELATION

It seems that the antinomic, paradoxical character of the situation was
first seen hv He explains the antinomy in its second form with

respect to an interchange of individual expressions as the first of the three
“puzzles” which he says every theory of denoting (name-relation) must
solve. He states the second form of the principle of interchangeability
(24-3b) in the following words: “If a is identical with b, whatever is true of
the one is true of the other, and either may be substituted for the other
in any proposition without altering the truth or falsehood of that proposi-
tion.”ii He takes as an example the sentence ‘ George IV wished to know
whether Scott was the author of Waverley’. If in this sentence, on the basis
of the true identity sentence ‘the author of Waverley is identical with
Scott’ (9-1), the description ‘the author of Waverley’ is replaced by
‘Scott’, the resulting sentence is presumably false.

Quine^^ likewise points out the second form of the antinomy with
respect to individual expressions. His first examples are psychological
sentences with the phrases ‘ is unaware that’ and ‘ believes that’ they are
similar to Frege’s example, ‘Copernicus asserts that , . (see above,
§ 28), and Russell’s example just mentioned. Further examples given by
Quine are modal sentences.^* The first is: ‘Necessarily, if there is life on
the evening star, then there is life on the evening star’. If here, on the
basis of the identity sentence, ‘The morning star is the same as the evening
star’ (28-2), which is found to be true by astronomical observations, one
occurrence of ‘the evening star’ is replaced by ‘the morning star’, a false
sentence results. (If, instead of the truth of the identity sentence 28-2, the
semantical statement 28-1 is used, we have the first form of the antinomy.)
In another example of a modal sentence, Quine uses numerical expres-
sions:

‘9 is necessarily greater than 7’.

If here, on the basis of the true identity sentence ‘The number of
planets = 9’, ‘9’ is replaced by ‘the number of planets’, the following
false sentence results:

‘The number of planets is necessarily greater than 7’.

I shall now give an example of the antinomy in both forms with respect
to predicators. We found earlier an ambiguity in the concept of the nomi-
natum of a predicator (for example, the German word ‘gross’ may be re-

[Denoting], p. 485.
Ibid.

[Notes], p. 1 15.

«Ibid.

Ibid., p. 121.

s 31. THE ANTINOMY OF THE NAME-RELATION 13 S

g^ed as a name of the class Large or of the property Large, see § 25).
Tn order to show that the antinomy of the name-relation is independent
of this ambiguity, the example will be formulated with phrases of the form
‘the class . . and only (^agses taken as nominata of these phrases. The
following sentence is true (‘necessaiy’ is here, as in earlier examples, used
in the sense of ‘logically necessary’):

^It is necessary that the class Featherless Biped is a subclass of the
class Biped’.

Now we replace in this sentence ‘ the class Featherless Biped’ by ‘ the class
Human’ ; this replacement may be based either, according to 24-3b, on the
circumstance that the identity sentence ‘the class Featherless Biped is the
same as the class Human’ is true (4-7) or, according to 24-3a, on the cir-
cumstance that the phrases ‘ the class Featherless Biped’ and ‘ the class
Human’ have the same nominatum. The result of the replacement is the
sentence

‘It is necessary that the class Human is a subclass of the class
Biped’.

Since, however, the fact that human beings have two legs is a contingent
biological fact and not logically necessary, the following is true:

‘It is not necessary that the class Human is a subclass of the class
Biped’.

The contradiction between these two results constitutes an instance of the
antinomy of the name-relation.

Those logical situations which are called logical antinomies (in the
modern, not the Kantian^nse) ofTo^cal paradoxes arfe^aracterized by
the fact that there are'wo methods of reasoning, which, although both
plausible and in accordance with customary ways of thinking, lead to
contradictory conclusion^Tflmy solution of ah antinomy, that is, the elimi-
nation of the contradiction, consists, therefore, in making suitable changes
in the reasoning procedure; at least one of its assumptions or rules must, in
spite of its plausibilityv-bC abolished or restricted in such a way ^at it
is no longer possible to reach the two incompatible conclusionsT’some-
times a certain form of inference is abolished or restricteJk^ometimes a
more radical step is taken by abandoning certain forms of ^ntences which
were previously regarded as meaningful and harmless!*1^us, for instance,
Russell’s solution of the antinomy known by his name consisted in the
rejection of sentences of the form ‘a«a’. Sometimes several different ways
for solving a given antinomy are found. It is a matter of theoretical

136

III. THE METHOD OF THE NAME-RELATION

investigation to discover the consequences to which each of the solutions
leads and, especially, what sacrifices of customary, and plausible ways of
expression or deduction each of them entailSi^ut which of the solutions
we choose for the construction of a language system is ultimately a matter
of practical decision, influenced, of course, by the results of the theoretical
investigation.

§ 32. Solutions of the Antinomy

Six procedures for the solution of the antinomy of the name-relation are dis
cussed. The first five still apply the method of the name-relation. Frege and
Church develop particular forms of this method by introducing certain dis-
tinctions, which, however, lead to a more complicated language. Russell re-
stricts to a considerable degree the application of the method of the name-
relation and thereby of the semantical analysis of the meaning of expressions.
Quine does the same to a smaller degreeV^he antinomy would also be elimi-
nated by restricting the language to extensional sentences^'Uut it is not known
at present whether the whole of logic and science is expressible in a language
of this kind.'PtnaUy, the method of extension and intension avoids the antinomy
by avoiding the concept of nominatumV^he concept of extension, though simi-
lar to that of nominatum, eliminates the contradiction without unnatural
restrictions and complications.

We shall now explain some of the solutions for the antinomy of the
name- relation which have been proposed or considered by logicians; we
call them Methods I-VI. First, we discuss five solutions which preserve
the method of the name-relation, at least to some extent. They may be
regarded as particular forms of this method. We shall find that each of
them has serious disadvantages. Then we shall consider the possibility of
solving the antinomy by giving up the method of the name-relation.

Method I, Frege. It seems that Frege was aware of the fact that the
principle of interchangeability (in the form 24 -^a) would lead to a con-
tradiction if the ordinary nominata of names were ascribed also to their
oblique occurrences and that the contradiction does not arise if different
nominata are ascribed to these occurrences.'in this sense we may say that
Frege offers a solution for the antinomy of the name-relation.tif' is true
that Frege does not speak explicitly of the necessity of avoiding a contra-
diction;^ gives other reasons for his distinction between the ordinary
nominatum and the oblique nominatum of a name.'^is reasoning gives
the impression that this distinction appeared to him natural in itself,
without regard to any possible contradiction. However, I think that to
many readers it will scarcely^pear very natural and that they, like my-
self, will see the strongesr^ument in favor of Frege’s method rather in
the fact that it is a way of solving the antinomy.

§32. SOLUTIONS OF THE ANTINOMY

137

yVhe disadvantages of Frege’s method have been explained earlier (§ 30).
■’We have seen that the unnecessary multiplicity of entities and names
which is generally a consequence of the method of the name-relation is
here even much greater. Furthermore, occurrences of the same name may
have different nominata — indeed, an infinite number of them; and in
certain contexts even the same occurrence of a name may have simul-
taneously several nominata.

Method n, Quine. Quine^’ uses the term ' designation’ for the name-
relation. He says of an occurrence of an expression in a nonextensional
context (as, for instance, ‘ the evening star’ in the first and ‘0’ ^the second
of the two examples of his, quoted in the preceding section) mat it is “ not
purely designative” and that it does not refer simply to the object
designated (the nominatum). He thinks that nonextensional contexts are
fundamentally different from extensional contexts ar^ more similar to
contexts in quotation marks; and, in particular, mat the customary
logical rules of specification and existential generalization are not valid for
nonextensional contexts (this will be discussed later, § 44). Thus his solu-
tion agrees with Frege’s in not ascribing the ordinary nominatum to an
occurrence of a name in a nonextensional context. But where Frege
ascribes a different nominatum, '^'uine ascribes no nominatum at all.
<!)onsequently, the principle of interchangeability (see his formulation
24-6) is declared by Quine not to be applicable to these occurrences, and
thus the antinomy is eliminated.

The advantage of Quine’s method in comparison with Frege’s consists
in avoiding the immense multiplication of entities and corresponding
names to which the latter method leads. But Quine’s method pays a high
price for this simplification by restricting the name-relation (‘designa-
tion’) 'W'^tensional contexts and grouping all nonextensional contexts
together with contexts in quotation marks and, furtherVGy imposii^ nar-
row restrictions upon the use of variables in' modal sentencesT^Those
logicians in particular who are interested in constructing or in semantical-
ly analyzing systems of modal logic will hardly be inclined tq adopt this
method.

Method III, Church. Chur^-’® regards Frege’s method as prefer-
able to Quine’s in two respects: first, because it provides that a name al-
ways has a nominatum’* even in nonextensional contexts and, second,
because^i«tf|;e’s conception of the sense of nan^ as something outside
the language (e.g., propositions or properties) seems more natural than
Quine’s way of construing the sense (meaning) of a name as ifs L-equiva-
[Notes]. J* [Review Q.j. i» Ibid., p. 46.

138 III. THE METHOD OF THE NAME-RELATION

lence class (see end of § 33). Howeyer, Church does not simply adopt
Frege’s method in its original form; he proposes important modifications
in it. He agrees with Frege’s conclusion that the nominatum of an oblique
(nonextensional) occurrence of a name must be different from its ordi-
nary nominatum and must be the same as its ordinary sense. But Church
seems to accept this only as a result of an'^alysis of nonextensional
•wntences as they occur in natural word languages and in syst^ns of modal
logic of the customary form. In a well-constructed language, however,
this multiplicity of nominata for the same name should be avoided.
Therefore, Church proposes, for semantical discussions in the natural
word languages, “to adopt some notational device to distinguish the
oblique use of a name from its ordinary use”;^“ this would be analogous
to the customary use of quotation marks. Mere distinguishing marks are
not sufficient, however, in a symbolic language system ; here we should go
one step further, as we do when we use not quotation marks but special
symbols as names of signs. “In a formalized logical system, a name would
be represented by a distinct symbol in its ordinary and in its oblique
use”.

I agree that, if the method of the name-relation is used, then the
changes in the notation proposed by Church are indeed an improvement.
On the other hand, it seems that these changes would cause an additional
complication in a system of modal logic. For example, there would be
an infinite number of types corresponding to the one type of sentences
in the method of extension and intension.

Although Church’s method avoids the multiplicity of nominata for the
same name, it shares the other complications of the original form of
Frege’s method explained in § 30. This fact, however, is not an argument
against Church’s method in comparison to the other forms of the method
of the name-relation. On the contrary, V^think that Church’s form of the
method may well be regarded as that which carries out the basic ideas
of the method of the name-relation in the most consistent and thorough
way, ehcjinating features not tolerable in a well-constructed system and
not restricting unduly the domain of application of the fundamental con-
cepts of the methodyTherefore, the great complications to which it leads
are to be regarded, rather, as an argument against the method of the
name-relation in general — provided that there is some other convenient
method which avoids them.

Method rV, Russell. Russelh* constructs the antinomy of the name-
relation with respect to individual expressions; in his example (see the
Ibid., p. 46. [Denoting].

i 32. SOLUTIONS OF THE ANTINOMY 139

pfeceding section) the description ‘the author of Waverley’ is replaced
by the proper name ‘Scott’. According to Russell’s conception, a descrip-
tion has no meaning in itself, but a sentence containing a description has
a meaning,^ and this meaning can be expressed without using the descrip-
tion. The contextual definition of a description (see above, § 7, Method
II) is a rule for transforming a sentence containing a description into a
sentence with the same meaning which no longer contains the description.
Although in the case of an individual description which fulfils the unique-
ness condition we may regard the one individual (the descriptum) as the
nominatum of the description, nevertheless, a sentence containing this
description is not about this individual. (Thus the principle of subject
matter, 24-2, is rejected with respect to descriptions.) What the sentence
actually means is shown only in its expanded form. Proper names (e.g.,
‘Walter Scott’) are regarded as abbreviations of description^<rlius, in the
primitive notation, neither proper names nor descriptions occur. There-
fore, the principle of interchangeability for individual expressions is not
applicable, and that form of the antinomy which arises from an inter-
change of individual expressions is eliminated. The situation is quite
analogous for abstraction expressions of classes (for example, ‘^(Ha:)’; see
the explanations above, at the beginning of § 26). These expressions are
likewise introduced by contextual definitions and not regarded as having
any meaning in themselves. The meaning of a sentence containing a class
expression is shown by its expansion in primitive notation, where no
class expression occurs. Thus, also with respect to class expressions the
principle of interchangeability is inapplicable, and the antinomy does not
appear.

If Russell regards sentences as names at all, then presumably he re-
gards them as names of propositions; in any case, he does not regard them
as names of truth-values. Thus the final result with respect to Russell’s
application of the name- relation may be summed up in the following way:
Although individual expressions and class expressions may, in a certain
sense, be regarded as naming individuals or classes, they do not occur in
the primitive notation but are incomplete symbols without independent
meaning)^ nominata in the strict sense, neither individuals nor classes
nor truth-values occur, hence none of those entities which we call exten-
sions. The antinomy of the name-relation arises from an interchange of
two expressions with the same nominatum. In all the chief kinds of in-
stances of the antinomy — including all instances mentioned in this book
and all instances given by the authors mentioned — the common nomina-

/Hd., p. 4S0.

14o

III. THE METHOD OF THE NAME-RELATION

turn is an extensionT^^ierefore, Kuril’s method, by excluding extensions
from the realm of nominata in the strict sense, eliminates at least the most
important instances of the antinomy.

A few remarks may be made on Russell’s objections to Frege’s method.
The chief objection^^ concerns the case of a description which does not
fulfil the uniqueness condition. Frege says that in this case the description
has a sense but no nominatum. Russell regards it as unsatisfactory that
expressions of the same syntactical form should in one case have a
nominatum and in another case not. Since, according to Frege, a sentence
is about the nominata of the expressions occurring in it (24-2), in the case
in which the uniqueness condition is not fulfilled the sentence is about no
entity at all; hence, Russell says,^'' one would suppose that the sentence
“ought to be nonsense; but it is not nonsense, since it is plainly false”.
This reasoning seems to me convincing; moreover, I suppose that Frege
himself would agree with it because he regards the feature mentioned as a
defect of natural languages.'*® This is the reason for his demand that in a
well-constructed language every description should have a nominatum by
virtue of a suitable convention."* Russell’s objection here is that this pro-
cedure is artificial and does not give an exact analysis of the actual use
of descriptions. However, Frege’s convention had a different purpose. He
first gave an analysis of the natural language and then proposed the con-
vention as a step not in the exact reconstruction of the natural language
but rather in the construction of a new language system intended to be
technically superior to the natural language.

Russell’s general objections"^ against Frege’s distinction between nomi-
natum and sense are rather obscure. This is due chiefly to Russell’s con-
fusion between use and mention of expressions, which has already been
criticized by Church."*

The disadvantage of Russell’s method lies in the fact that meaning is
denied to individual expressions and class expressions. That these kinds of
expressions can be introduced by contextual definitions and hence that
what is said with their help can also be said without them is certainly a
result of greatest importance but does not seem a sufl&cient justification
for excluding these expressions from the domain of semantical meaning
analysis. It must be admitted, I think, that descriptions and class expres-
sions do not possess a meaning of the highest degree of independence; but
that holds also for all other kinds of expressions except sentences (see re-

Ibid., p. 41; see above, § 8.

Op. cit., pp. 485-88.

<* (Review C.], p. 302.

« Ibid., pp. 483 f.
« Ibid., p. 484.
[Sinn], p. 40.

§ 32 . SOLUTIONS OF THE ANTINOMY 141

marks at the end of §1). And it is certainly useful for the semantical
analysis of the meanings of sentences to apply that analysis also to the
meanings, however derivative, of the other expressions, in order to show
how out of them the independent meanings of the sentences are con-
stituted.

Method V, Extensional Language. The most radical method for
eliminating any antinomy arising in connection with certain forms of ex-
pression consists in excluding these forms entirely. In the case of the
antinomy of the name-relation this solution would consist in excluding
all nonextensional contexts^^ other words, in using a purely extensional
language (see the definition ii-2c). To construct an extensional language
system for certain restricted purposes involves, of course, no difficulties
(as examples of such systems, see, e.g., Quine’s language system ML and
my systems I and II in [Syntax]). But this is not sufficient for the present
purpose. In order to eliminate the antinomy by excluding all nonexten-
sional contexts, i^ 1 ^Id be necessary to show that for the pTIrposes of
any logical or empirical field of investigation an extensional language
system can be constructed; in other words, that for any nonextensional
system there is an extensional system into which the former can be
translated. The assertion to this effect is known as \htHfiesis of extension-
alUy.*’’ The problem of whether it holds or not is still unsolved. Translat-
ability into extensional sentences has been shown for certain kinds of non-
extensional sentences. Thus, for instance, any simple modal sentence is
L-equivalent to a semantical sentence in an extensional metalanguage
using L-terms, as we shall see later (§ 39).®“ For example, the modal sen-
tence ‘N(A)’, in words: ‘it is necessary that A’, is L-equivalent to the
semantical sentence ‘ ‘A’ is L-true’ (according to a convention to be dis-
cussed later). The application of this method of translation to sentences
with iterated modalities (e.g., ‘it is necessary that it is possible that . . .’)
involves a certain difficulty; this, however, can be overcome, as I have
shown at another place.®" The translation of nonextensional sentences
with psychological terms like ‘believes’, ‘knows’, etc., is presumably like-
wise possible, although at present it is not yet clear how it can best be

See [Syntax], § 67; [I], p. 249; Russell, [Inquiry], chap. xix.

S” For this translation see [Syntax], § 69; I would now define the L-concepts not as syn-
tactical but as semantical concepts (see above, § 2). Note that in this translation the two
sentences, although L-equivalent, are not intensionally isomorphic (§ 14). A translation in the
stronger sense, preserviirg intensional structure, is obviously impossible between a nonexten-
sional and an extensional sentence.

S’ [Modalities].

142

lU. THE METHOD OF THE NAME-RELATION

made (see the discussions in §§ 13 and 15). The question of whether an
extensional language is sufficient for the purposes of semantics will be dis-
cussed later (§ 38) ; an affirmative answer does not seem implausible, but
the question is not yet definitely settled.

If we could prove the thesis of extensionality and if we decided to ex-
clude all nonextensional sentence forms, then obviously the antinomy of
the name-relation would be eliminated. Furthermore, the difference be-
tween the method of the name-relation and the method of extension and
intension would disappear, since, with respect to extensional occurrences,
the nominatum of an expression is the same as its extension, and its sense
the same as its intension (29-1 and 2). Attractive though these conse-
quences may appear, it seems to me that it would be at least premature to
propose Method V as a solution of the antinomy at the present time.
Even if the thesis of extensionality were proved, this would not be suf-
ficient as a justification for Method V. We should have to show, in addi-
tion, tha^n e 5 rtensional language for the whole of iJs^ and sctS^e is not
only possible but also technltfally more efficient than nonextensional forms
of language. Though extensional sentences follow simpler rules of deduc-
tion than nonextensional ones, a nonex^Sn^onal language often supplies
simpler forms of expression ; consequently, even the deductive manipula-
tion of a nonextensional sentence is often simpler than that of the compli-
cated extensional sentence into whiclftf would be translated. Thus both
forms of language have their advantages; and the problem of where the
greater over-all simplicity and efficiency is to be found is still in the bal-
ance. Much more investigation of nonextensional, and especially of modal,
language systems will have to be done before this problem can be decided.
Therefore, for the time being, Method V as a solution of the antinomy has
to be left aside.

Method VI, Extension and Intension. If, instead of the method of the
name-relation, the method of extension and intension is used for semanti-
cal analysis, then tlieconc^t nfri ominatiim d oes no t occpr, and hence the
antinomy of the name-relation in its original form cannot arise, ^inre^

ho^ver^ the concept of extension is in many rpsppcts pnH part- ^

Iv coincides with the concept of nom inatum. there mig ht arise, un der cer-
tain conditions, an antinomy of the identity of extension analogous toTlRtt-^

of the identity of nominatum. The antinomy would arise if for the concept
of extension a principle analogous to the principle of interchangeability of
names (24-3) were laid down. The form which we have chosen for the
method of extension and intension excluJles the anttfiomy by prescribing

§ 32. SOLUTIONS OF THE ANTINOMY

143

for expressions with the same extension, in other words, for equivalent
expressions, a principle of interchangeability which is restricted to exten-
sional contexts (12-1). Our second principle (12-2) concerns L-equivalent
expressions, hence those with the same intension; thus it is related to
Frege’s second principle (28-9).

Perhaps it will occur to the reader at this point to ask why, if a restric-
tion of interchangeability to extensional contex^s^sures the elimination
of the antinomy, we might not simply keep Frege’s two concepts and re-
strict his first prin^le to extensional (nonoblique) contexts. The reply is
that Frege’s concept (‘bezeichnen’) is meant in the sense of a name-rela-
tion, that is, as a relation characterized by the principles 24-1 and 2;
therefore it would be quite implausible and unnatural, as we have seen
earlier, not to maintain the principle of interchangeability 24-3 in its unre-
stricted form. Or, to put it the other way round, if somebody uses a con-
cept for which the principle 24-3 does not hold unrestrictedly, then this
concept is not a name-relation and is not the concept meant by Frege
and many other logicians, for example, Church and Quine's^

It is easy to see that the method of extension and intension avoids those
features of the other methods which we have found to be disadvantages.
In our general discussion of the method of the name-relation, Mre have first
explained the ambiguity in the concept of nominatum (§25); for in-
stance, even if we understand clearly what is meant by a given predicator,
we may regard either the property or the class as its nominatum. The con-
cept of extension does not involve any analogous ambiguity; the exten-
sion of any predicator of level one and degree one is thaclass of tho se in di-
viduals to which the predicator can be truly applied>Turther, we have
shown the multiplicity of expressions in the object language to which the
method of the name-relation leads (§ 26) ; we have seen tha^f our meth-
od is used, this multiplicity is replaced by one expressionVirurther, the
complications caused by the particular form of the method introduced by
Frege have been explained (§30). Their common root is the fact that
different occurrences of the same expression may have different nominata.
Since the extension of an expression is always the same, independNit of
the context, no analogous complications are caused by our methoorThe
disadvantage of Quine’s method is the restriction of the name-relation to
extensional contexts; there is no apalogous restriction of the ap-
plication of the concept of extensioncAVTule Church’s method avoids
some of the disadvantages of Frege’s original method, it shares most of
them; further, his modification of Frege’s method, necessary though it is.

144 in. THE METHOD OF THE NAME-RELATION

causes a new complication, which does not occur in our method.^l^e dis-
advantage of Russell’s method is its denial of meaning to individual ex-
pressions and class expressions. In our method there is no such restriction;
to every expression of these kinds an extension and an intension are
ascribed (for class expressions in the system PM, see above, § 26).

Let us sum up the result of our discussion of the method of the name-
relation in this chapter. The method appears in various forms with dif-
ferent authors. Most authors who use the concept of the name-relation do
not seem to be aware of the antinomy and do not develop the method in a
sufficiently explicit form to enable us to see whether and how they avoid
the contradiction. All procedures that have been proposed for the elimi-
nation of the antinomy have serious disadvantages; some of these pro-
cedures lead to great complications, others restrict considerably the field
of application of the semantical meaning analysis. Thus it seems doubtful
whether the method of the name-relation is a suitable method for se-
mantical analysis.

CHAPTER IV

ON METALANGUAGES FOR SEMANTICS

In the metalanguage M, which we have used so far, we have spoken about
extensions and intensions, for instance, about classes and properties. It is the
main purpose of this chapter to show that this distinction does not actually
presuppose two kinds of entities but is merely a distinction between two ways
of speaking. First, we discuss possible methods for defining extensions in terms
of intensions or vice versa, without adopting any of them (§ 33). Then we con-
struct a new metalanguage M' (§§34-36). While M contains distinct expres-
sions for an extension (e.g., 'the class Human’) and an intension (e.g., ‘the
property Human’), M' contains only one expression (e.g., ‘Human’), which is,
so to speak, neutral, like the expressions in the symbolic system S, (e.g,, ‘H’).
Therefore, we call M' a neidral metalanguage. By this elimination of the dupli-
cation of expressions, the apparent duplication of entities disappears. It is
shown that all sentences of M can be translated into M', including the semantics
of systems like S, (§ 37). Finally, the question is examined as to whether a com-
, plete semantical description of a system, even a nonextensional system like S3,
can be formulated in a metalanguage which, in distinction to M and M', is
extensional; it seems that this is the case (§ 38).

§ 33. The Problem of a Reduction of the Entities

In the metalanguage M we have so far spoken as if there were two kinds of
entities in each type, extensions and intensions, for example, classes and
properties. Here the question is discussed as to whether we can get rid of this
apparent duplication of entities by defining one kind in terms of the other. Four
methods for defining extensions in terms of intensions are discussed; the concep-
tion of extensions as L-determinate intensions (§23); Russell’s contextual
definition of classes in terms of properties, which is shown to involve a certain
difficulty; a modified version of Russell’s definition, which avoids the difficulty;
and, finally, a method which uses properU- expressions themselves as class
expressions but presupposes a particular structure of the language. It does not
seem possible to define intensions themselves in terms of extensions. However,
the class of all designators L-equivalent to a given designator might be taken as
a representative of its intension.

We have used as metalanguage M a part of English, modified and
supplemented in a certain way (§ i). Throughout our discussions we have
used in M terms like ‘class’, ‘property’, ‘truth- value’, ‘proposition’, ‘indi-
vidual’, ‘individual concept’, and the more general terms ‘extension’ and
‘intension’. This manner of speaking gives the appearance of dealing
with a great variety of entities and, in particular, with two kinds of enti-
ties within each type. As stated at the beginning (§ 4) , we have used the
terms mentioned only because they help to facilitate understanding, but

145

146 IV. ON METALANGUAGES FOR SEMANTICS

our theory is not based on the assumption that there are entities of all
these kinds. Now, mindful of Occam’s razor, we shall try to show how the
number of apparent entities can be reduced to half. Since the apparent
duplication of entities was actually only a duplication of terminology, all
we have to do is to construct another way of speaking which avoids the
terminological split into extensions and intensions.

Let us begin with the discussion of predicators, because here the dis-
tinction between extension and intension is customary and familiar. If we
wish to have a language which is not, like Si, restricted to elementary
statements about things but contains a more comprehensive system of
logic and especially of mathematics, then we must introduce means for
speaking in general terms not only about things but also about entities of
higher levels, say classes or properties. So much is admitted even by those
logicians who are most wary in admitting abstract entities.' The question
is whether it is necessary to admit both kinds of entities, classes and
properties, or whether those of the one kind are definable with the help of
those of the other. For instance, is one of the two phrases (in M) ‘ the class
Human’ and ‘ the property Human’ definable with the help of the other?
Explicit definition is not necessary; a contextual definition would sufl&ce
to make one of the two phrases dispensable in the primitive formulation.

Let us first look for methods which define class expressions in terms of
property expressions.

1. If the concept of L-determinate intensions (§ 22) is available, we
can define ‘the class /’ as ‘the L-determinate property which is equivalent
to the property /’ (§ 23).

2. If we do not wish to make use of the concept of L-determinate in-
tensions, we may consider the possibility of a contextual definition for ‘ the
class /’ by a generalized reference to the properties which are equivalent to
the property /. Since all these properties determine the same class, the
most natural procedure seems to be to interpret a statement about the
class / as a statement about all these properties. Thus, for a system S,
containing predicate variables etc., we could lay down the following
contextual definition for the class-expression ^x{fx)’\

33 - 1 . ‘..^(/rr)..’ for ‘(g) [(g^/) D ..g..]’.

This definition must be supplemented by a rule specifying what is to be
taken, in any given case in which ‘ f (/r)’ occurs, as the context ‘ . . ^(Jx) . .’

' See, for example, Quine, [Notes], p. 125; “Anyone who cares to explore the foundations
of mathematics must, whatever his private ontological dogma, begin with a provisional toler-
ance of classes or attributes [i.e., properties)”.

§ 33. THE PROBLEM OF A REDUCTION OF THE ENTITIES 147

to which the definition is to be applied. Following Quine* rather than
Russell (see below), we stipulate that the definition is to be applied to the
smallest sentence or matrix in the primitive notation in which the class
expression occurs. Thus, before applying the definition, we have to trans-
form the given sentence containing a class expression by eliminating all
previously defined signs with the help of their definitions; then, with the
help of definition 33-1, we expand each smallest matrix in which the class
expression occurs.

3. Russell’ was the first to propose a contextual definition of class ex-
pressions on the basis of property expressions. Whitehead and Russell
used this definition in their construction of the system of mathematics in
[P.M.].^ Though the method has been able to supply a good working
basis for this construction, there is one feature of the definition which
seems to me disadvantageous. The definition given above (33-1) is, in-
deed, nothing else than a variant of Russell’s definition, changed, however,
with respect to the point in question. The definition in [P.M.], transcribed
in our notation,’ is as follows:

33 - 2 . ‘. . Kfz) . for‘( 3 g)[(g ^ f) . . . g .

The definiens here contains an existential quantifier, not a universal one,
as does 33-1. Thus a statement about the class / is here interpreted as a
statement not about all properties but about at least one property equiva-
lent to the property / (in the terminology of [P.M.], “at least one proposi-
tional function formally equivalent to the propositional function
Russell does not explain his reasons for the form of the definition chosen,
except for saying, correctly, that the definiens ought to be extensional;
this, however, is likewise the case if a universal quantifier is used, as in
33 - 1 -

The form of the definition with the existential quantifier seems to me not
only to be less natural but also to lead to serious difficulties, which make

» (M.L.], § 26.

> Russell, “Mathematical Logic as Based on the Theory of Types”, American Journal of
Mathematics, XXX (1908), 222-62; for the definition, see p. 249.

qP.M.], I, 71 a., 187 a.

® The definition in the original notation (p. 249 of the article mentioned above, and [P.M.],
I, 76, i88) is as follows:

*20 oi./fzC^s)} . = ; 4 >\x . Df

Our transcription 33-2 is changed in inessential respects only. The exclamation point is omitted
because it is necessary only on the basis of the ramified system of types, which is now generally
regarded as unnecessary, and because it is at any rate inessential for the problem under dis-
cussion. The context is indicated only by dots instead of by a second-level variable, in order to
make the definition applicable also to systems not containing such variables. The biconditional
sign is used according to our abbreviation 3-1.

148

IV. ON METALANGUAGES FOR SEMANTICS

it appear doubtful whether the definition fulfils the purpose intended. To
show this, let us consider two nonextensional properties of properties, say,

and such that is the contradictory of ; hence ^2 holds in all
cases, and only in those, in which <i', does not hold. Since is nonexten-
sional, there are different, but equivalent, properties, say f, and fi, such
that #, holds for f, and not for fj, and hence ^2 holds for fj. Then, accord-
ing to definition 33-2, both 4 >, and hold for the class z(iiz), although

and €>2 are contradictories and hence logically incompatible. This would
be an awkward result, although it does not constitute a formal contra-
diction, since and #2 are logically exclusive only with respect to proper-
ties, while their application to classes is introduced merely as a certain
mode of speech, which in the formal system itself, as distinguished from
the informal interpretation in terms of classes, is merely a device of ab-
breviation.

In order to see the situation more clearly, let us try to construct a con-
crete example. As earlier (§ 26), let PM be the system constructed in
[P.M.], and PM' be the same system with some nonlogical constants
added on the basis of rule 25-1. In order to find something like ^>i and
$2 in PM or PM', we have to look for nonextensional signs. Among the
very few such signs occurring in the system PM itself, there are the signs
of identity ‘ = ’ and nonidentity ‘ ’ when standing either between

property expressions or between a property expression and a class expres-
sion. The sign ‘ = ’ is actually used in [P.M.] in this way;*" and the authors
are aware that it is nonextensional in these contexts.^ We shall first use the
system PM'. We take as premises the following two sentences of this
system :*

(i) ‘(2:)(Fx • Bx = Hx)’, or briefly, ‘F • B = H’.

(ii) ‘FS.Bz 5^ m’.

These sentences say that the property Featherless Biped and the property
Human are equivalent but not identical. Hence they are true. Now we
shall examine the following two sentences;

(iii) ‘g(Hz) = Hz’.

(iv) ‘g(Hz) HI’.

"^See [P.M.], I, igi, the proofs of *20.13 and *20.14.

’ Ibid., p. 84.

* For the convenience of the reader, we transcribe the notation of [P.W.] into our notation
by writing the quantifier in the form ‘(x)’ instead of in the form of a subscript and by using
parentheses instead of dots. We keep, however, the notation ‘Hz’ for a property expression be-
cause this is an essential feature of the notation in (P.M.l (see above, § 26).

§ 33. THE PROBLEM OF A REDUCTION OF THE ENTITIES 149

We shall expand these sentences by applying Russell’s definition 33-2 in
order to eliminate the class expression ‘S(Hz)’. We substitute in this defi-
nition ‘H’ for'/’; as‘. . l(Hz) . we take (iii) and (iv) in turn. Thus (iii)
is expanded into

(v) ‘(ag)[a - H) . (g 2 = m)]’.

This sentence is provable in PM', because it follows by existential general-
ization from the instance with ‘H’ for ‘g’. Therefore, (iii) is provable and
hence true on the basis of the interpretation assumed (§26). Now let us
expand (iv). Here we have to take into consideration Russell’s rule of con-
text, according to which the smallest sentence or matrix in the actually
given abbreviated notation is to be taken as corresponding to the left
side in the definition 33-2. In other words, ‘ is not to be eliminated be-
fore the elimination of the class expression, and hence the whole of (iv)
is to be taken as ‘ . z(H2) . Thus we obtain as the expansion of (iv) :

(vi) ‘( 3 g)[(« - H) . (g 2 ^ m)]\

This sentence is derivable from the conjunction of our premises (i) and
(ii) by existential generalization with respect to ‘F 5 • Bl’. Hence, (iv) is
derivable from the premises and therefore likewise true. Thus the result
is that the sentences (iii) and (iv) are both true, although they look like
contradictories. They do not actually constitute a contradiction because
(iv) is not meant as the negation of (iii) ; this is shown by the fact that,
according to the rules of the system PM', (iv) is expanded not into
‘~(z(Hz) = Hz)’ but into (vi). Nevertheless, our result shows that the
notation of the system PM' is here misleading, because it suggests the
interpretation of (iv) as“z(Hz) is not identical with HS”, which would be
in contradiction to (iii). It is true that Russell warns repeatedly that the
class expressions are incomplete and have no meaning in isolation. On the
other hand, the notation has been constructed with this aim in mind;
The class expressions should be such that they can be manipulated as if
they were names of entities; and Russell seems to assume that this aim
has been reached." Our result makes this assumption doubtful.

In the system PM itself, without the use of nonlogical constants, we
can reach a similar result. Here we take as premise the assumption that
there are two properties which are equivalent but nonidentical. Any par-

’ It is stated ([P.M.], p. 188) that with regard to the scope of class expressions the same
conventions are adopted as for descriptions. That the sign ‘ ^ ’ when occurring in combination
with a description is not eliminated before the elimination of the description is seen from the
example in [P.M.], p. 173, line 2 from bottom.

” [P.M.], p. 188, 11 . 3-5 and 14-16; and the text of p. 198.

IV. ON METALANGUAGES FOR SEMANTICS

ISO

ticular instance — for example, the conjunction of (i) and (ii) — can be
formulated only in PM', not in PM. But the existential assumption can
be formulated in PM itself as follows;

(vii) ‘( 3 ^)(a/)[(g-/) .(^2^/2)]’.

In a way similar to the above we can derive from this premise in PM the
following:

(viii) ‘( 3 f)[(B(fz) =fB) • (S(fz)

This sentence is not provable in PM, but it is derivable from the premise
(vii), which is, no doubt, true on the basis of the interpretation intended
in [P.M.J; this work itself mentions the example of the properties Feather-
less Biped and Human. Although (viii) is not actually self-contradictory,
still it looks as if it were. This shows again that the way the class expres-
sions are introduced by Russell’s definition is not quite in agreement with
the intended purpose.

If, instead of Russell’s definition 33-2, a definition involving a universal
quantifier like 33-1 is used, then (iii) is not provable. In this case, both
(iii) and (iv) are false. This apparently, but not actually, violates the
principle of excluded middle; however, this seems less disturbing than the
previous apparent violation of the principle of contradiction. If, further-
more, the rule of the context of the class expression is changed from
Russell’s form (the smallest sentence in the actually given abbreviated
notation) to Quine’s form (the smallest sentence in the primitive notation),
as was done in 33-1, then (iv) is expanded into the negation of (iii). In this
case, (iii) is false and (iv) is true, and thus there are no longer any puzzles. If
in [P.M.] the definition of classes were changed according to 33-1, then
only some of the proofs in a few subsections referring to the definition
would need to be changed. It seems that later only extensional contexts
occur; therefore, the theorems and proofs throughout the bulk of the work
would remain unchanged.

4. Suppose that the language system S in question is such that every
smallest matrix, that is, one which does not contain another matrix as a
proper part, is extensional. This is the case, for instance, if modal operators
are the only nonextensional signs. [Therefore, it is the case in Sa, where
‘N’ is the only nonextensional sign. Here, every nonextensional matrix
contains a (proper or improper) part of the form ‘N(. . .)’ and hence a
matrix ‘ . . .’ as a proper part. On the other hand, it is not the case in the
system PM' if we take it as including the sign ‘ = ’ between property
expressions. For example, the sentence ‘Hf = Hf’ is of smallest size, but
it is intensional (see above).] Then every class expression in S stands, after

§ 33 . THE PROBLEM OF A REDUCTION OF THE ENTITIES 151

the elimination of all other defined signs, within a smallest matrix which
is extensional. Therefore, the class expression can here simply be replaced
by the corresponding property expression, even if the smallest matrix in
question stands within a wider nonextensional context. [For example,
‘N[a€£(. . X . .)]’ or ‘N[i6(. . x . .)(a)]’ is L-equivalent to, and hence L-
interchangeable with, ‘N[(Xa:)(. . x . .)(a)]’, and hence also with
‘N(. . a . The reason for this is as follows. Let the smallest matrix
containing a certain occurrence of ‘^(Ha;)’ be represented by
This is, according to our definition 33-1, L-equivalent to ‘ (g) [ (^ = H) D
This obviously L-implies but the latter also L-implies the
former (ra-i), since, according to our assumption with respect to S,
is extensional. Therefore, the two sentences are L-equivalent, and hence
also L-interchangeable even in intensional contexts (12-2).

This shows that, in a system S of the kind described, we may simply
take the property expressions themselves as class expressions also. This
procedure is still simpler than procedure (2) explained above, which uses
the contextual definition 33-1 for class expressions.

We have discussed four methods for the definition of classes in terms of
properties. They can be used more generally for the definition of exten-
sions of any kind in terms of intensions. These methods as here explained
apply to symbolic object languages. The same methods can, of course, be
applied in an analogous way to a word language and, in particular, to our
metalanguage M. This latter application would be more important for us,
because in our symbolic object languages we do not want to have class
expressions in addition to property expressions, for the reasons explained
earlier (§ 26), while in M we have phrases of both forms'the class Human’
and ‘ the property Human’ and we should like to dispense with one of
these forms in the primitive formulations in M. Since M contains identity
sentences for properties (like PM'), it does not fulfil the condition re-
quired for 5 in method (4). But we could apply method (2) to M. This
would consist in laying down the following three definitions; the first cor-
responds to 33-1, the second and the third are analogous to it:

33 - 3 . . . . the class / . . . = of for every property g equivalent to the
property /, . . . the property g ... .

33 - 4 . . . . the truth- value p . . ■ = of for every proposition q equivalent
to the proposition />,... the proposition q ... .

33 - 6 . . . . the individual x . . . = di for every individual concept y equiva-
lent to the individual concept x, . . . the individual concept y ... .

152

IV. ON METALANGUAGES FOR SEMANTICS

A Convention determining the context indicated by dots would here be
laid down similar to that for 33-f. (We may disregard here inessential
changes of this context required by the accidents of idiom; for example,
‘x belongs to the class/’ is changed to 'x has the property /’.)

The three definitions here mentioned will not actually be adopted for
M, because we shall find another, simpler form of a metalanguage which
avoids even the apparent duplication of entities in M by entirely avoiding
the duplication of expressions. This will be explained in the next section.

Would it be better to take properties as primitive and to define classes in
terms of properties or to take classes as primitive and to define properties
in terms of classes? We have explained four methods for the first alterna-
tive. Quine” rejects it for the reason that a prop)erty is even more obscure
than a class. Which of the two is more obscure and which intuitively
clearer is a controversial question. I shall not discuss this question here; it
seems to be more psychological than logical. However, I think that most
logicians agree that, if the terms ‘ class’ and ‘ property’ are understood
in their customary sense, classes can be defined by properties, but it is
hardly possible to define properties by classes (unless these classes are, in
turn, characterized by properties); for a property determines its class
uniquely, while many properties may correspond to a given class. It is,
however, possible to define in terms of classes certain entities which stand
in a one-one correlation to properties or other intensions and therefore
may represent them for many purposes. We defined earlier the L-equiva-
lence class of a designator in 5 as the class of all designators in S L-equiva-
lent to it (3-1 5b). It is easily seen that there is a one-one correlation be-
tween the L-equivalence classes in S and the intensions expressible in S.
Therefore, the L-equivalence class of a designator in 5 may be taken as
its intension or at least as a representative for its intension. Procedures of
this kind have been indicated by Russell and by Quine. Russell” mentions
as a possibility the definition of a proposition as“ the class of all sentences
having the same significance as a given sentence”. Quine’’ defines the
meaning of an expression as the class of those expressions which are
synonymous with it. Russell’s concept of having the same significance and
Quine’s concept of synonymity correspond at least approximately to our
concept of L-equivalence; if a stronger relation than L-equivalence is
meant, for example, something like intensional isomorphism (§ 14), the
concepts are, of course, analogous.

[Notes], p. 126.

“ [Inquiry], p. 209.

[Notes], p. 1 20.

1 34. THE NEUTRAL METALANGUAGE M'

1S3

§ 34. Hie Neutral Metalanguage M'

While some symbolic systems (e.g., Russell’s) have different expressions for
properties and for classes, our systems (S, and Sa) have only one kind of expres-
sion. Analogously, we now introduce a “neutral” metalanguage M'. While M
contains phrases like ‘the property Human’ and ‘the class Human’, M' con-
tains only the neutral expression ‘Human’; and similarly with other types of
designators. In this way the duplication of expressions in M is eliminated in
M', and thus the apparent duplication of entities disappears.

If, of the two phrases ‘ the class Human’ and ‘ the property Human’ in
M, either the first were defined by the second or vice versa, then in the
primitive notation of M we should have only one phrase instead of two,
and hence the number of entities would be cut in half. I think that the
same aim can also be reached in another and even simpler way. We have
seen earlier (§ 26) that, on the basis of the method of extension and inten-
sion, the notation in a symbolic object language can be simplified. Instead
of one expression as a name of a property (e.g., ‘Hf’ in PM') and another
expression as a name of the corresponding class (e.g., ‘^(Ha:)’), it is suf-
ficient to use one expression (e.g., ‘ (Xa;) (Ha:) ’ or ‘H’ in Si). This expression
is, so to speak, neutral in the sense that it is regarded neither as a name of
the property nor as a name of the class but rather as an expression whose
intension is the property and whose extension is the class. If we apply an
analogous procedure to the word language M, then our aim will be at-
tained. Thus we have to look for a language form M' in which we use, in-
stead of the two phrases ‘the class Human’ and ‘the property Human’,
only one phrase; this phrase, however, is not to be one of the two but
rather another one which is neutral in containing neither the word ‘ class’
nor the word ‘property’. The simplest procedure is to take the word
‘human’ or ‘Human’ alone (the capitalized form to be used, as previously,
at places where English grammar does not permit an ordinary adjective).
We take M' as the neutral metalanguage which results from M by these
changes, that is, by eliminating the terms ‘class’, ‘property’, etc., in favor
of neutral formulations. Our tasks is now to find suitable forms for formu-
lations in M'. In this discussion we shall speak about M and M', and hence
we shall speak in a metametalanguage MM. For easier understanding, we
take MM similar to M rather than to the less familiar M' ; that is to say,
we shall use terms like ‘class’, ‘property’, ‘extension’, ‘intension’, etc., in
speaking about M', although these terms cannot occur in M' itself. The
very next sentence will, in fact, be an example of this use. The term
‘Human’ in M' is neutral in the same sense in which ‘H’ is neutral in Si:
‘Human’ is regarded neither as a name of a class nor as a name of a prop-

1S4

IV. ON METALANGUAGES FOR SEMANTICS

erty; it is, so to speak, at once a class expression and a property expression
in the following way :

34-1. The extension of ‘Human’ in M' is the class Human.

34-2. The intension of ‘Human’ in M' is the property Human.

Analogously, instead of the two phrases ‘ the individual Scott’ and ‘ the
individual concept Scott’ in M, we have in M' the one neutral term' Scott’.
Here we have :

34-3. The extension of ‘ Scott’ in M' is the individual Scott.

34-4. The intension of ‘Scott’ in M' is the individual concept Scott.

Since classes and properties have different identity conditions, a diffi-
culty arises in the translation of identity sentences into M'. Take as an
example the following sentences in M (see § 4) :

34-6. The class Human is the same as the class Featherless Biped.

34-6. The property Human is not the same as the property Featherless
Biped.

34-7. The property Human is the same as the property Rational Animal.

We translated, above, two phrases in M into ‘Human’ by simply omitting
the words ‘the class’ and ‘the property’. However, if we were to do the
same with 34-5 and 34-6, a contradiction would obviously result. Gen-
erally speaking, since identity is different for extensions and intensions, a
neutral formulation cannot speak about identity. Hence, identity phrases
like ‘is identical with’ or ‘is the same as’ are not admissible in M'. How,
then, to translate identity sentences into M'? Here the terms ‘equivalent’
and ‘L-equi valent’ in their nonsemantical use, as defined by 5-3 and 5-4,
will help; note that in this use the terms stand for relations, not between
designators, but between intensions. The definitions show that identity of
extensions coincides with equivalence of intensions, and identity of inten-
sions coincides with L-equivalence of intensions. Here in M', the terms
‘equivalent’ and ‘ L-equivalent’ can be used in connection with neutral
phrases instead of phrases for intensions without any difficulty; therefore,
we shall speak of equivalence and L-equivalence of neutral entities. Thus
the general rules for the translation of identity sentences (in M or in a
non-neutral object language, e.g., PM') into neutral formulations in M'
are as follows:

34-8. A sentence stating identity of extensions is translated into M' as a
sentence stating equivalence of neutral entities.

34-9. A sentence stating identity of intensions is translated into M' as a
sentence stating L-equivalence of neutral entitles.

§ 34 . THE NEUTRAL METALANGUAGE M' 155

Accordingly, we translate the identity sentences 34-5, 34-6, and 34-7 in M
into the following sentences in M';

34 - 10 . Human is equivalent to Featherless Biped.

34 - 11 . Human is not L-equivalent to Featherless Biped.

34 - 12 . Human is L-equivalent to Rational Animal.

These three sentences can be obtained from 5-5, 5-6, and 5-7 in M by
simply dropping the phrase ‘the property’.

The sentences 34-10, 34-11, and 34-12 must be clearly distinguished
from the following sentences, which look similar but are fundamentally
different in their nature :

‘Human’ is equivalent to ‘Featherless Biped’ in M'.

‘Human’ is not L-equivalent to ‘ Featherless Biped’ in M'.

‘Human’ is L-equivalent to ‘Rational Animal’ in M'.

These sentences are semantical sentences in MM concerning certain predi-
cators in M'. Therefore, the predicators are included in quotation marks,
and the sentences contain references to the language M'. They are perfect
analogues to the sentences 3-8 and 3-1 1, which are semantical sentences
in M (or M') concerning predicators in S,. On the other hand, the sen-
tences 34-10, 34-11, and 34-12 are not semantical sentences; they do not
speak about the predicators but use the predicators in order to speak
about nonlinguistic entities. Therefore, the predicators are not included
here in quotation marks, and there is no reference to a language system.
The sentences belong to the nonsemantical (and, moreover, to the non-
semiotical) part of M', to that part into which the sentences of the object
languages can be translated. Sentence 34-10 is not only a translation of
the sentence 34-5 in M, but also of the corresponding identity sentence
26-9 in PM' and in ML' (§ 25) ; 34-10 is, furthermore, an exact translation
of the likewise neutral sentence ‘H s F • B’ of S,. Since 34-11 and 34-12
are intensional (in the sense of 11 -3b), there cannot be sentences exactly
corresponding to them in the extensional language S, (§ 1 1 , Example IV) .
But there are such sentences in the modal language S2, as we shall see
later; thus‘H = RA’ corresponds to 34-12.

Now let us see how neutral formulations of sentences are to be framed
in M'. The translation of simple sentences, especially atomic sentences,
into M' involves no difficulty, since it corresponds closely to the custom-
ary formulation. Thus, for instance, as a translation of ‘Hs’ we take
in M' the simplest of the translations in M, namely, ‘Scott is human’
(which is 4-1). The other translations into M earlier discussed are not
neutral (for instance, 4-2, 4-3, and those with ‘ individual’ and ‘individual

iS6 IV. ON METALANGUAGES FOR SEMANTICS

concept’ analogous to those mentioned in § 9) ; hence they are excluded
from M'. The neutral formulation, ‘ Scott is human’, in M' replaces not
only the non-neutral sentences in M just mentioned but also the two non-
neutral phrases, ‘ the truth- value that Scott is human’ and ‘ the proposition
that Scott is human’ in M (see 6-3 and. 6-4). In some cases the simple
formulation ‘Scott is human’ does not comply with ordinary English
grammar, for instance, when occurring as a grammatical subject. In these
cases we might, in analogy to ‘Human’, capitalize all words: ‘Scott-Is-
Human’; but this would be rather awkward for longer sentences. Another
alternative is the addition of ‘ that’ (see remark on 6-3 and 6-4) : ‘ that Scott
is human’. This formulation is to be used only as part of larger sentences,
especially in the translation of sentences of M containing one of the
phrases ‘the truth- value that Scott is human’ or ‘the proposition that
Scott is human’. In some cases, this formulation agrees with ordinary
usage, in others not; but we shall admit it into M' in all cases. Thus the
(false) sentence 'N(Hs)’ (in Sj) is the translated into 'It is necessary that
Scott is human’. Since ‘Hs’ is equivalent to ‘(F • B)s’, the following is
true in M':

34 - 13 . That Scott is human is equivalent to that Scott is a featherless
biped.

This formulation is admittedly somewhat awkward. The more customary
formulations in M with ‘the proposition’ or ‘the truth-value’ inserted
after ‘to’ are not possible here in M' because they are not neutral; and
there is no customary neutral noun. Therefore, we decide to admit the
form 34-13 in M', and likewise the analogous form 34-14 below.

Since ‘Hs’ is L-equi valent to ‘RAs’, the following is true in M':

34 - 14 . That Scott is human is L-equivalent to that Scott is a rational
animal.

The use of ‘equivalent’ and ‘L-equivalent’ as nonsemantical terms stand-
ing between sentences, as in 34-13 and 34-14, is in analogy to the use of
these terms between predicators (as in 34-10, etc.) and individual expres-
sions, but here, between sentences, it is still more at odds with ordinary
grammar. Fortunately, there is another formulation which is customary
and grammatically correct; but it has the disadvantage that it is ap-
plicable only in connection with sentences, not with other designators.
Instead of ‘equivalent’ we may use here ‘if and only if’, and instead of

‘L-equivalent’ ‘that ... if and only if , is necessary’. (Here the

phrase ‘is necessary’ is placed at the end only for the reason that English
provides no other simple means to indicate that the argument of this

§35. M' IS NOT POORER THAN M 157

phrase is the whole ‘if and only if’ sentence and not only its first compo-
nent.) In this way, the following sentences take the place of 34-13 and

34-14:

34-16. Scott is human if and only if Scott is a featherless biped.

34-16. That Scott is human if and only if Scott is a rational animal, is
necessary.

§ 36. M’ Is Not Poorer than M

The question is raised as to whether the designators in M' are correctly
described as neutral or whether they are, perhaps, actually names of intensions
in disguise. If somebody wishes to regard ‘Human’ in M' (or ‘H’ in S3) as the
name of a property, there is no essential objection. But it would be wrong to
say that a language like Sa or M' contains only names of properties and no names
of classes and therefore lacks important means of ejqiression. Actually, all
sentences of M are translatable into M'. That M' is not poorer in means of
expression than M is also shown by the possibility of reintroducing the non-
neutral formulations of M into M' with the help of contextual definitions.

Perhaps a reader who is accustomed to the usual method of the name-
relation will have some doubts as to whether the language M' or any other
language can possibly be genuinely neutral; he will say that the allegedly
neutral word ‘Human’ in M', and likewise the corresponding sign ‘H’ in
Si and S3, in order to be unambiguous, must mean either as much as ‘the
propjerty Human’ or as ‘the class Human’; in other words, it must be a
name either of the property or of the class (compare Quine’s comments
below, § 44). I cannot quite agree with this either-or formulation. I think
we should rather say that the word ‘Human’, and likewise ‘H’, stand
both for the property as its intension and for the class as its extension.
However, it must be admitted that the neutrality is not quite sym-
metrical. As we have seen earlier (§ 27), a designator' stands primarily for
its intension ; the intension is what is actually conveyed by the designator
from the speaker to the listener, it is what the listener understands. The
reference to the extension, on the other hand, is secondary; the extension
concerns the location of application of the designator, so that, in general,
it cannot be determined by the listener merely on the basis of his under-
standing of the designator, but only with the help of factual knowledge.
Therefore, if somebody insists on regarding a designator as a name either
of its intension or of its extension, then the first would be more adequate,
especially with respect to intensional languages like M' and S,. I think
there is no essential objection against an application of the name-relation
to the extent just described, for example, against regarding ‘Human’ in
M' and ‘H’ in Sa as names of the property Human. The only reason I

iS8 IV. ON METALANGUAGES FOR SEMANTICS

would prefer not to use the name-;^lation even here is the danger that this
use might mislead us to the next step, which is no longer unobjectionable.
In accordance with the customary conception of the name-relation, we
might be tempted to say: “If ‘Human’ (or ‘H’) is a name for the property
Human, where do we find a name for the class Human? We wish to speak,
not only about properties, but also about classes; therefore, we are not
satisfied with a language like M' or Sj, which does not provide names for
classes and other extensions.” This I should regard as a misconception of
the situation. M' is not poorer than M by not containing the phrase ‘the
class Human’. Whatever is expressed in M with the help of this phrase is
translatable into M' with the help of ‘Human’; and whatever is expressed
in a non-neutral symbolic language like PM' with the help of the class
expression ‘:^(Ha;)’ is translatable into Sj with the help of ‘H’. The simplest
method for the translation into S, is based on the method (4) explained in
the preceding section. For the sake of an example, let us take, not the
system PM', but the system PM'', which is like PM' except for containing
our form of the contextual definition of classes (33-1) instead of Russell’s
(33-2)- The rules of translation from PM'' into S2 with respect to class
expressions are as follows:

35 - 1 . a. For the translation of a smallest sentence (or matrix) which is
extensional and does not contain ‘ = ’, both a property expression
(e.g., ‘H^) and a class expression (e.g., in PM" are

translated into Sj by the corresponding neutral expression (e.g.,
‘H’ or ‘ (Xa:) (Ha;) ’). (This rule is based on method (4), explained
in § 33-)

b. An identity sentence in PM" with two class expressions (e.g.,
‘^(Ha;) = f (Fa: • Ba;)’) is translated into an =-sentence with
the corresponding neutral expressions (e.g., ‘H = F • B’).

c. An identity sentence with two property expressions (e.g.,
‘H:^ = RA^’) is translated into the corresponding = -sentence
(e.g.,‘H = RA’).

(We leave aside here identity sentences with one class expression
and one property expression; all such sentences are L-false.)

The translation from M into M' is analogous. We may assume that any
sentence of smallest size in M which is not an identity sentence is exten-
sional. Then the rules are as follows:

35 - 2 . a. In all contexts except identity sentences, both class expressions
(e.g., ‘the class Human’) and property expressions (e.g., ‘the

§ 35 . M' IS NOT POORER THAN M 159

property Human’) are translated by the corresponding neutral
expressions (e.g., ‘Human’).

b. A sentence stating the identity of classes is translated into a
sentence stating the equivalence of the corresponding neutral
entities.

c. A sentence stating the identity of properties is translated into a
sentence stating the L-equivalence of the corresponding neutral
entities.

Rules 35-2b and c are special cases of the general rules 34-8 and 34-9 for
extensions and intensions (see examples 34-10, 34-11, and 34-12).

Thus we see that the view that M' is poorer than M is a misconception.
Since the formulation “The designators in M' are names for intensions,
and there are no names for extensions in M'”, may easily lead to this mis-
conception, it seems to me inadvisable. It seems more adequate and less
misleading to say either “every designator in M' has an intension and an
extension” or “the designators in M' are neutral”.

In the translations by rule 35-24, characterizing phrases like ‘the class’,
‘the property’, etc., are simply dropped. This might give the impression,
perhaps, that in the transition from M to M' certain important distinc-
tions disappear. This, however, is not the case. All the distinctions made
in M are preserved in M'; they are only formulated in a different and, in
general, in a simpler way. This is shown by the fact that all the non-
neutral ways of speaking in M with terms like ‘class’, ‘property’, etc.,
could be reintroduced into M' by contextual definitions if we wanted them
there. (In fact, of course, we do not want to destroy the neutrality of
M'.) Thus the terras ‘class’ and ‘ property’ could be introduced by the fol-
lowing contextual definitions:

36 - 3 . a. . . . the class / . . . =Df for every g, if g is equivalent to/, then

. . . g . . . .

b. . . . the property / . . . =Df for every g, if g is L-equivalent to/,
then ... g ... .

(Concerning the context indicated by dots, see the remarks following
33-5.) If the context indicated by dots is extensional, we may take, in-
stead of (a), the simpler definition:

a'. ... the class /...= or .../.. . .

If the context is either extensional or intensional, we may take, instead of
(b), the simpler form:

b'. . . . the property / ... =Df •■./.. . .

i6o IV. ON METALANGUAGES FOR SEMANTICS

Contextual definitions for the terms ‘individual’, ‘individual concept’,
‘truth-value’, and ‘proposition’ are analogous. It is admitted that these
definitions lead in some cases to unusual formulations. However, they do
not lead to false results. The decisive point is that they also yield the origi-
nal non-neutral formulations in M.

Identity sentences like those in M can likewise be reintroduced into M'
by a procedure the reverse of that described in 34-8 and 34-9:

36-4. a. The class / is the same as the class g = of / is equivalent to g.
b. The property / is the same as the property g =Df/isL-equivalent
tog.

The possibility of these definitions in M' for the non-neutral formula-
tions in M shows that all distinctions in M are actually preserved in M'
in a different form. In other words, M' is not poorer in means of expression
than M.

§ 36. Neutral Variables in M'

Some symbolic systems have different variables for classes and for properties;
we have seen earlier (§ l^) that this is unnecessary. Similarly, the phrases
‘for every class’ and ‘for every property’ in M constitute an unnecessary dupli-
cation. They are replaced in M' by ‘for every f, where ‘/' is a neutral variable
whose value-intensions are properties and whose value-extensions are classes.
Neutral variables for other types are introduced analogously.

There are still other non-neutral expressions in M which have to be re-
placed by neutral expressions in M', namely, those phrases by which we
refer in a general way to entities of some kind, for instance, pronouns
like ‘every’, ‘any’, ‘all’, ‘some’, ‘there is’, ‘none’, in combination with
words like ‘class’, ‘property’, etc. In a symbolic language, phrases of this
kind are translated with the help of variables in quantifiers. We have seen
earlier (§ 27) that in a symbolic language not only the use of different ex-
pressions for classes and for properties is an unnecessary duplication, but
so is likewise the use of different variables for classes and for properties
(as, for instance, ‘a’ and ‘<ft’ in the system PM). Instead, we may use
neutral variables, whose value-extensions are classes and whose value-
intensions are properties. Now we shall do the same in M', in order to
make possible the neutral formulation of general sentences. We supple-
ment the word language in M' by the following letters as variables: ‘/’,
‘g’, etc., for predicators of level one and degree one as value expressions;
‘x’, ‘y’, etc., for individual expressions; ‘p\ ‘y’, etc., for sentences. Thus, a
non-neutral formulation of a universal sentence in M containing one of
the two phrases ‘every class’ (or ‘all classes’) and ‘every property’ (or ‘all

§ 37. SEMANTICS IN THE NEUTRAL METALANGUAGE M' i6i

properties’) is translated into a neutral sentence of M' with the help of
the phrase ‘for every f, corresponding to a universal quantifier. Likewise,
an existential sentence in M containing one of the phrases ‘ some class’ (or
‘ there is a class’) and ‘ some property’ (or ‘ there is a property’) is translated
into M' with the help of ‘for some/’ (or ‘there is an/’). Analogously, a
general sentence concerning propositions or truth-values is translated
into M' with the help of ‘for every p’ or ‘for some p’ (or ‘ there is a p’). And
a general sentence concerning individuals or individual concepts is trans-
lated with the help of ‘for eveiy a:’ or ‘for some x’ (or ‘there is an x’).
Examples will be given later (see 43-4).

Universal sentences in M about extensions or intensions in general can
likewise be translated into M' if we introduce general variables for which
designators of all types are value expressions. To avoid contradictions,
suitable restrictive rules have to be laid down for the use of these general
variables; this can be done in different ways.'^

§ 37. On the Formulation of Semantics in the Neutral Metalanguage M'

Two semantical relations between expressions and neutral entities, designa-
tion and L-designation, are introduced into M'. It is shown how semantical
rules and statements in M can then be translated into M'. The relation of
designation is extensional; it is used for the translation of statements concerning
the extension of given expressions. The relation of L-designation is nonexten-
sional ; it serves for the translation of statements concerning the intension of
expressions. Thus the whole semantics of a system (e.g., S,) can be translated
from M into M'.

In the preceding sections we have discussed only the nonsemantical
part of the metalanguage, that part into which the sentences of object
languages can be translated. We come now to the more important
semantical part of the metalanguage, that part in which we speak about
the sentences and other expressions of the object languages, applying to
them semantical terms like ‘true’, ‘L-true’, ‘equivalent’, ‘L-equivalent’,
etc. Most of the discussions in the earlier chapters of this book are formu-
lated in this semantical part of the metalanguage M. This holds, in par- ,
ticular, for those statements which speak about classes, properties, prop-
ositions, etc., not only in a general way but in relation to expressions of
an object language — for instance, the following two (§ 4) :

37-1. The extension of ‘H’ in S, is the class Human.

37-2. The intension of ‘H’ in S, is the property Human.

” For a historical survey of different methods of avoiding the antinomies, see Quine,
[M.L.], § 29.

i 63 IV. ON METALANGUAGES FOR SEMANTICS

The important question now is whether it is possible also to translate
these semantical statements of M into the neutral metalanguage M', that
is to say, into formulations which, instead of phrases like ‘ the class Hu-
man’ and ‘the property Human’, use only neutral phrases like ‘Human’.
Only if this is possible can we say that we have overcome the duplication
of entities.

We shall see that it is indeed possible to translate semantics from M
into M'. The sentence 37-1 states that the relation of extension holds be-
tween the class Human and the predicator ‘H’ (in S,), and 37-2 states
that the relation of intension holds between the property Human and the
same predicator. How can we obtain neutral formulations in M' referring
to the neutral entity Hiunan instead of to the class and the property?
It would, of course, not do simply to drop the phrases ‘ the class’ and ‘ the
property’ in those sentences, because then the same entity would be as-
serted to be at once the extension and the intension of the same predica-
tor, and that would not be in accordance with the intended meaning of
the terms ‘extension’ and ‘intension’. Instead, we must make use of a
relation which holds between the neutral entity Human and the predica-
tor ‘H’, a relation which can be neither the relation of extension nor that
of intension, although it is similar to them. A closer investigation of the
situation shows that we need here two new relations, both holding between
‘H’ and Human; the first of them is related to the second as a radical
semantical concept (e.g., truth) to the corresponding L-concept (e.g.,
L- truth). Therefore, it seems natural, if we find a suitable word for the
first relation, to take the same word with the prefix ‘L-’ for the second.
The first relation is here meant in such a way that it is definable also in an
extensional metalanguage; but the second relation is intensional, as we
shall see. Since the first relation holds between an expression (e.g., ‘H’)
and an entity (e.g.. Human) for which that expression stands, a word like
‘ means’, ‘ signifies’, ‘ expresses’, ‘ designates’, ‘ denotes’, or something similar
would seem suitable. I do not wish to make a specific suggestion. Let us
tentatively use the term ‘designates’ for the first relation, and hence
‘ L-designates’ for the second. Then, instead of 37-1 and 37-2 in M, we
have in M', with respect to S„ the following:

37 - 3 . ‘H’ designates Human.

This may be regarded as the formulation in M' of a rule of designation for
the system S, (corresponding to the first item in the previous rule 1-2).
The first relation is intended to be extensional; that is to say, any full
sentence of it is extensional with respect to each of the two argument

§ 37 . SEMANTICS IN THE NEUTRAL METALANGUAGE M' 163

expressions. Hence, 37-3 is extensional with respect to ‘Human’; that is
to say, the occurrence of ‘ Human’ in this sentence is interchangeable with
any predicator which is equivalent to ‘Human’ in M'. Thus we obtain the
following two results, according to the equivalences stated in § 34:

37 - 4 . ‘H’ designates Featherless Biped.

37 - 5 . ‘H’ designates Rational Animal.

By using a neutral predicator variable ‘/’ (see § 36) and ‘equivalent’ as
a nonsemantical term (see 5-3 and § 34), we can express the result in a
general form :

37 - 6 . For every /, if / is equivalent to Human, then ‘ H’ designates /
(in S,).

If a suitable definition for ‘designates in Si’ is laid down, which has not
been done here, then the converse of 37-6 also holds :

37 - 7 . For every /, ‘H’ designates / (in S,) if and only if / is equivalent to
Human.

We have decided to use the term‘L-designates’ for the second relation.
We shall not give a definition for it. We assume for the following discus-
sion that it is defined with respect to a given system, say Si, in such a way
that the following condition 37-8 is fulfilled; an analogous condition holds
for L-truth, according to our convention 2-1, and for the other L-concepts.

37 - 8 . An expression 21 . L-designates an entity « in Si if and only if it can
be shown that 21, designates u in Si by merely using the semantical rules
of S„ without any reference to facts.

(The variable ‘ m’ here used in M' is a general, that is, not type-restricted,
variable; see the remarks at the end of § 36.) Now let us apply 37-8 to
37-3, 37-4, and 37-s in turn. Statement 37-3 can be established on the basis
of the semantical rules of S, alone in a trivial way, since it is itself one of
these rules. This yields, with respect to S, :

37 - 9 . ‘H’ L-designates Human.

The same, however, does not hold for 37-4. In order to show that this
statement holds, we have used and must use not only the semantical rule
37-3 but also the result that the predicators ‘Human’ and ‘Featherless
Biped’ are equivalent in M'; this equivalence, like that of the correspond-
ing predicators ‘H’ and ‘F • B’ in Si, is not an L-equivalence (see § 34)
but is based on biological fact (3-6). Hence, according to 37-8, the follow-
ing is true in M';

37 - 10 . ‘H’ does not L-designate Featherless Biped.

i 64 IV. ON METALANGUAGES FOR SEMANTICS

Since 'Human' and ‘Featherless Biped’ are equivalent in M', we see from
37-9 and 37-10 that the relation of L-designation is nonextensional.

Statement 37-5 can again be established on the basis of rule 37-3 alone,
without reference to facts, because ‘ Human’ and ‘ Rational Animal’ are
supposed to mean the same (see remark on 1-2). Hence, according to 37-8,
the following is true;

37 - 11 . ‘H’ L-designates Rational Animal.

We can formulate the result in a general form with a neutral variable ‘/’
and ‘ L-equivalent’ as a nonsemantical term :

37 - 12 . For every /, if / is L-equivalent to Human, then ‘ H’ L-designates/.

If a suitable definition for ‘L-designates’ is laid down in accordance with
the convention 37-8, then also the converse of 37-12 holds:

37 - 13 . For every /, ‘H’ L-designates / if and only if / is L-equivalent to
Human.

Statement 37-3 maybe regarded as a translation of 37-1 into M',and like-
wise 37-9 as a translation of 37-2. It is true that the explicit reference to a
class in 37-1 is not directly mirrored by any expression in 37-3, but it is
indirectly represented by the extensionality of 37-3 with respect to ‘Hu-
man’, which is shown by the instance 37-4 and generally by 37-6. Thus,
37-6 may also be regarded as a translation of 37-1. Similarly, the explicit
reference to a property in 37-2 is indirectly represented by the intensional-
ity of 37-9 with respect to ‘Human’, which is exhibited in instances like
37-10 and 37-11 and generally in 37-12. Thus, 37-12 may also be regarded
as a translation of 37-2.

We have shown the application of the relations of designation and L-
designation to predicators. The application to designators of other types is
quite analogous. As examples with respect to individual expressions in S„
in analogy to 37-3, 37-4, 37-6, and 37-7, the following sentences are true
in M':

37 - 14 . ‘s’ designates Walter Scott.

37 - 16 . ‘s’ designates The Author Of Waverley.

37 - 16 . For every x, if x is equivalent to Walter Scott, then ‘ s’ designates x.
37 - 17 . For every 2:, ‘s’ designates x if and only if * is equivalent to Walter
Scott.

Rule 37-14 is a rule of designation of the system S,, corresponding to
the first item in 1-1. Sentence 37-15 is derived from 37-14 with the help
of a historical fact (9-1). Further, in analogy to 37-9, 37-10, 37-12, and
37-13, the following sentences are true in M':

§ 37. SEMANTICS IN THE NEUTRAL METALANGUAGE M' 165

37-18. ‘s’ L-designates Walter Scott.

37-19. ‘s’ does not L-designate The Author Of Waverley.

37-20. For every x, if x is L-equivalent to Walter Scott, then ‘s’ L-desig-
nates x.

37-21. For every x, ‘s’ L-designates x if and only if * is L-equivalent to
Walter Scott.

Sentences 37-i4and37-i 6 maybe regarded as translations of the follow-
ing sentence in M (§ 9) :

‘The extension of ‘s’ is the individual Walter Scott’.

Sentences 37-18 and 37-20 may be regarded as translations of:

‘The intension of ‘s’ is the individual concept Water Scott’.

Remarks analogous to those made above on 37-3 and 37-9 hold here.

Analogously, with respect to sentences in S„ the following statements
are true in M'; we use a that-clause for the neutral formulation (§ 34):

37-22. ‘Hs’ designates that Scott is human.

This statement, in distinction to 37-3 and 37-14, is itself not a semantical
rule but follows from these rules with the help of a suitable definition for
‘designates in Si’, as applied to sentences. The following is a consequence
of 37-22, because ‘Scott is human’ and ‘Scott is a featherless biped’ are
equivalent in M':

37-23. ‘Hs’ designates that Scott is a featherless biped.

Generally, with the neutral variable ‘p’ (§ 36):

37-24. For every p, if p is equivalent to that Scott is human, then ‘Hs’
designates p.

37-26. For every p, ‘Hs’ designates p if and only if p is equivalent to that
Scott is human.

(In these two statements, the nonidiomatic phrase ‘is equivalent to that’
may be replaced by ‘if and only if’; see the explanations to 34-13 and
34-IS-)

Furthermore, for L-designation, the following statements are true in
M':

37-26. ‘Hs’ L-designates that Scott is human.

37-27. ‘Hs’ does not L-designate that Scott is a featherless biped.

37-28. For every p, if p is L-equivalent to that Scott is hiunan, then
‘Hs’ L-designates p.

i66

IV. ON METALANGUAGES FOR SEMANTICS

37 - 29 . For every p, ‘Hs’ L-designates p if and only if p is L-equivalent to
that Scott is human.

(In the last two statements, the nonidiomatic phrase ‘is L-equivalent to
that’ can be avoided by a transformation analogous to that of 34-14 into
34-16.)

Sentences 37-22 and 37-24 may be regarded as translations of the sen-
tence 6-3 in M concerning the truth-value as extension; likewise, 37-26
and 37-28 as translations of the sentence 6-4 concerning the proposition as
intension. Remarks analogous to the earlier ones hold here.

We have previously seen that it would be possible to reintroduce the
non-neutral terms ‘class’, ‘property’, etc. into M’ by contextual defini-
tions. If we were to apply these terms in the formulation of semantical
statements in M', these statements would become quite similar to those
in M. For example, by applying the definition of ‘class’ (35-3a) to 37-6,
we obtain :

37 - 30 . ‘H’ designates the class Human.

Likewise, by applying the definition of ‘property’ (3S-3b) to 37-12, we
obtain:

37 - 31 . ‘H’ L-designates the property Human.

Analogous results would be obtained for individual expressions and
sentences. These results show that the relation of designation in M' cor-
responds to the relation between a designator and its extension in M, and
the relation of L-designation in M' corresponds to the relation between
a designator and its intension in M.*’

' ' My use of the terms ‘designation’ and ‘designatum’ in [I] was, as I realize now, not quite
uniform, because at that time 1 did not yet see clearly the distinction which I make now in M
with the help of the terms ‘extension’ and ‘intension’, and in M' with the help of the terms
‘designation’ and ‘L-designation’. The use of ‘designatum’ in [ 1 ] corresponds in most cases to
the present use of ‘intension’ in M (or ‘L-designatum’ in M'). Thus, in the Table of Designata
([I], p. 18) and in later examples of Rules of Designation, the following kinds of entities are
taken as designata: properties, relations, attributes, functions, concepts, and propositions. It
is only with respect to individual expressions that I used the term in a different way, taking as
designata in the table and in the examples not individual concepts but individuals. Since it is
not customary to speak of individual concepts under any term, I was not aware of the fact
that they, and not individuals, belong to the same category as properties, propositions, etc.
Thus, in the case of individual expressions, what I took as designata were the same as what
would be taken as nominata by the method of the name-relation. It is probably due to this fact
that Church ([Review C.]) understood ray term ‘designatum’ in all cases in the sense of
‘nominatum’; and presumably Quine ([Notes]) likewise believes himself to be in accord with
my use when he applies ‘designatum’ in this sense, I regret that the lack of a clear explanation
in [II has caused these misunderstandings. This lack was not accidental but was caused by an
obscurity of long standing in some of the fundamental semantical concepts. If I see it correctly,
this obscurity has been overcome only by the analysis made in this book. Church’s statement
((Review C.], pp. 299 f.) that the designatum of a sentence is not a proposition but a truth-
value is — on the basis of Frege’s method of the name-relation — correct for Qiurch’s use of

§ 37. SEMANTICS IN THE NEUTRAL METALANGUAGE M' 167

The examples in this section show how semantical sentences in M, stat-
ing the extensions or intensions of predicators, individual expressions,
and sentences in Si, can be translated into neutral formulations in M'.
The translation of semantical sentences which refer not to nonlinguistic
entities but only to expressions in the object language, for instance, sen-
tences about truth, L-truth, equivalence, and L-equivalence, does, of
course, not involve any difficulty. Thus the whole of semantics, with re-
spect to Si or any other system, can be translated from M into M'.

The reasons for our use of the two metalanguages, M and M', may be
briefly summarized. Metalanguage M was used in the first three chapters
of this book in an uncritical way, so to speak. It supplies pairs of terms
'class’-' property’, and the like, and the general terms ‘extension’ and ‘in-
tension’. The use of these terms constituted what we have called the meth-
od of extension and intension. The chief reason for using these pairs of
terms is that they correspond to familiar concepts, usually regarded as
kinds of entities. In the present chapter we constructed the neutral meta-
language M', which has no such pairs of terms and thus avoids the appear-
ance of a duplication of entities. Although the terms ‘extension’ and ‘in-
tension’ do not occur in M', the essential features of the method used in
M' are still the same as in M; therefore, we might still call the method
used in M' the neutral form of the method of extension and intension, or
else the (neutral) method of equivalence and L-equivalence, or the
(neutral) method of designation and L-designation. The distinctions made
in M are not neglected in M' but are represented in a different form. In-
stead of an apparent duplication of entities, we have here a distinction
between two relations among expressions, namely, equivalence and L-
equivalence, and, based upon it, a distinction between two relations be-
tween expressions and entities, namely, designation and L-designation.
We have seen that it is possible to construct in M' contextual definitions
for the non-neutral terms ‘class’, ‘property’, etc., which lead to formula-
tions like those in M. This result shows, on the one hand, that the neutral
method in M' does indeed preserve all distinctions originally made in M
and hence is an effective substitute for the original form of the method. On
the other hand, the result is a justification for M, since it shows that the

‘designatum’ in the sense of ‘nominatum’; not, however, for my use of ‘designatum’ in [I] in
the sense of ‘intension’.

In [I], I occasionally used the terms ‘synonymous’ and ‘L-synonymous’. The distinction
which I had in mind but did not grasp satisfactorily is now expressed more adequately by the
terms ‘equivalent’ and ‘L-equivalent’ in their application to designators in general.

i68

IV. ON METALANGUAGES FOR SEMANTICS

apparent duplication of entities in M is, in fact, only a duplication of
modes of speech.

Since the non-neutral mode of speech in M and the neutral mode of
speech in M' cover the same domain, the choice between them is a matter
of practical preference. The neutral formulation is much simpler and
avoids even the appearance of a duplication of entities. Therefore, this
formulation might be preferable in cases in which a metalanguage for
semantical purposes is to be constructed in a strict, systematized way, for
instance, in a symbolic language or in words whose use is regulated by
explicit rules. On the other hand, the non-neutral formulation is in most
cases more familiar, more in accordance with ordinary usage. Therefore,
this formulation may seem preferable for semantical discussions which
are not on a highly technical level, especially for purposes of introductory
explanations. That is the reason for its use in the first part of this book.

§ 38. On the Possibility of an Extensional Metalanguage for Semantics

The question is discussed as to whether a complete semantical description
of a system, even a nonextensional system like Sj, can be formulated in an
extensional metalanguage, for instance, the sublanguage Me of M' contain-
ing only the extensional sentences of M'. It is found that most of the semantical
rules (rules of formation, of truth, and of ranges) can be formulated in M,
without any difficulty. The situation is not so simple with respect to the rules
of designation; but it seems that these rules can also be adequately formulated
in Mg.

We have formulated semantical sentences in two different metalan-
guages, M and M'. Both these languages are nonextensional. The question
arises as to whether semantics can be formulated in an extensional meta-
language — more exactly, whether it is possible to construct an extensional
metalanguage sufficient for the formulation of a complete semantical de-
scription even of a nonextensional object language (as, for instance, SJ.
A semantical description of an object language is complete if it, given as
the only information about the language, enables us to understand every
sentence of the language and hence to determine whether or not it is
L-equivalent to any given sentence of our metalanguage. The answer to
the question is not at present known. However, on the basis of some
studies I have made, an affirmative answer seems to me not improbable.
Here I shall give a few indications only.

It is easily seen that a sentence in M which says what the intension of a
certain expression is, is nonextensional. For example, the sentence ‘the
intension of ‘H’ in Si is the property Human’ (4-17) is nonextensional with
reject to ‘ the property Human’, because if this predicator is replaced by

f 38. AN EXTENSIONAL METALANGUAGE FOR SEMANTICS 169

the equivalent one, ‘the property Featherless Biped’, then the true sen-
tence is changed into a false one. Sentences of this kind are essential for
the use of our method in M. Therefore, if we wish to find extensional
semantical sentences, it seems more promising to look at the neutral
formulations in M'. The term 'intension' does not occur in M'; nor do
those intensional sentences of M which state the identity or the non-
identity of properties or other intensions (for instance, 4-8 and 4-9).
Nevertheless, M' is not extensional; the semantical formulations which
we used in M' contain the following three nonextensional (and, moreover,
intensional) terms and no others. The first is the modal term ‘necessary’
(see, for instance, 34-16). The second is the term ‘L-equivalent’ in its
nonsemantical use, as occurring, for instance, in 34-11, 34-12, and 34-14;
it is easily seen that each of these sentences is nonextensional with respect
to both argument expressions. This term is definable on the basis of
‘necessary’ (compare, for instance, 34-1-4 and 34-16). [Note, incidentally,
that the semantical term ‘L-equivalent in the system S' is extensional.
For example, ‘ ©i is L-equivalent to in the system S,’ is extensional; in
contradistinction to 34-14, it does not contain sentences as parts, but only
names of sentences.] The third nonextensional term in M' is ‘ L-designates’
(see the remark following 37-10).

Let Me be the metalanguage which contains all the extensional sen-
tences of M' and no others; we can construct it out of M' by omitting all
sentences containing the three nonextensional terms mentioned. Our ques-
tion is: How much of the semantics, say of the extensional system S, and
the intensional system S^, can be formulated in M^?

A complete system of semantical rules for S, or S2, which is not given in
this book, would consist of the following kinds of rules:

(i) Rules of formation, on the basis of a classification of the signs;
these rules constitute a definition of ‘sentence’.

(ii) Rules of designation for the primitive descriptive constants,
namely, individual constants and predicates.

(iii) Rules of truth.

(iv) Rules of ranges.

It is easy to see that the rules of kinds (i), (iii), and (iv) can be formu-
lated in an extensional metalanguage like M^. We must here consider these
rules in their exact formulation. The designations of expressions of the
object language must be formed, not with the help of quotation marks, as
we did for the sake of convenience in the previous examples of semantical
rules and statements, but as descriptions with the help of German letters.

170 IV. ON METALANGUAGES FOR SEMANTICS

Let us add here, for this purpose, the letter 'iR' as designation in M, of the
modal sign ‘ N’ in S,. As an example of a rule of formation for Sj in M,, let
us take the rule for N-matrices: ‘If 21. is a matrix in Sa, then 92(21.) is a
matrix in S,. ’ In application to the instance ‘ Hs’ , this rule says that, if ‘ Hs’
is a matrix, as it is, indeed, according to another rule, then ‘N(Hs)’ is a
matrix. Note, however, that the rule itself does not contain the expres-
sion 2l„ for instance, ‘Hs’, but only refers to this expression by using a
name ‘21,’ for it (actually, a variable for which a name, say ‘2li’, may be
substituted). Among the rules of truth we leave aside for the moment that
for atomic sentences because it contains the term ‘designates’ (or ‘refers
to’, see 1-3) which will be discussed later. The following is an example of
one of the other rules of truth (1-5): ‘A disjunction of two sentences ©,■
and ©> (that is to say, a sentence consisting of ©, included in parentheses
followed by the wedge followed by ©,• included in parentheses) is true if
and only if either ©i or ©, or both are true.’ It is clear that this formula-
tion is extensional. The same holds for the rules of ranges for S,, which
will be given in § 41 . These rules define ‘ the sentence ©, holds in the state-
description is a class of sentences. Note that the sentence ©„

let alone the class does not itself occur in the rule; only the names (or
variables) ‘ ©,’ and ‘ ®„’ occur. Thus it is clear that the relation of holding
is extensional. The rules of ranges refer, moreover, to assignments; an as-
signment is a function which assigns to a variable and a state-description
as arguments an individual constant as value. Only the extensions of
these functions are essential for the rules and the statements based upon
the rules; that is to say, if a reference to one assignment in a true state-
ment is replaced by a reference to another equivalent assignment (i.e.,
one which assigns to all pairs of arguments the same values as the first
assignment), then the resulting statement is likewise true. Note, further,
that the exact formulation of the rule concerning‘N’ (4i-2g) has the form:
‘A matrix 92(2li) holds . . .’; thus it does not contain the modal sign ‘N’
itself but only its name ‘ 92 ’. Thus we see that all rules of ranges for S2,
including the rule concerning ‘N’, are extensional.

Now we go back to the rules of designation. Here is the one critical
point for our problem of the expressibility of the semantics of S, in M,.
In M', we distinguished two relations between designators and neutral
entities, namely, designation and L-designation. The relation of designa-
tion is extensional and hence does occur in M,; but the relation of L-
designation does not. Thus we have to examine the question as to whether
the relation of designation suffices for describing the meanings of the ex-
pressions in the object languages. For instance, the meaning of ‘H’ in S,

538. AN EXTENSIONAL METALANGUAGE FOR SEMANTICS 171

and Sa is (the property) Human, not Featherless Biped; the meaning of
‘s’ is Walter Scott, not The Author Of Waverley. In M', we can easily
express this distinction with the help of the term ‘L-designation’ by the
statements 37-9 and 37-10, 37-18 and 37-19. But how can we do it in M„
where we have only the term ‘designation’? The difficulty consists in the
fact that, with respect to designation, the following two statements are
both true (37-3 and 37-4) :

38 - 1 . ‘H’ designates Human.

38 - 2 . ‘H’ designates Featherless Biped.

And the same holds for the following two statements (37-14 and 37-15):

38 - 3 . ‘s’ designates Walter Scott.

38 - 4 . ‘s’ designates The Author Of Waverley.

In view of this fact, it might seem at first glance as though it were impos-
sible to give in M, the information about the meanings intended for ‘H’
and ‘s’. However, I believe that this is not impossible. In M, we lay down
38-1 and 38-3 among the rules of designation for S, and S3. Then the state-
ment 38-2, although it is likewise true, is fundamentally different from
38-1, for it is neither a semantical rule, nor derivable from the semantical
rules alone; it was derived from rule 38-1 together with a biological fact
(3-6). If the metametalanguage MM, in which we are speaking here about
M, and the other metalanguages, contains L-terms, then we can formu-
late the difference in this way: 38-1 is L-true in M, but 38-2 is only F-true.
The relation between 38-3 and 38-4 is analogous. But even in M, itself we
can describe the situation in more explicit terms. If we wish to add to 38-1
a negative statement in M^, the following may be taken (3-8) :

38 - 6 . ‘H’ and ‘F • B’ are not L-equivalent (in S, and Sa).

This statement, together with 38-1 and some other semantical rules, cor-
responds in a certain sense to the negative statement 37-10 in M'.

The rules of designation themselves refer only to the primitive indi-
vidual constants and predicator constants. But the extensional relation of
designation can also be defined in M, in a wider sense so as to apply to all.
designators, including compound individual expressions, predicators, and
sentences, also intensional sentences in S3. Then, for example, the fol-
lowing two statements hold in Me (37-22 and 37-23):

38 - 8 . ‘Hs’ designates that Scott is human.

38 - 7 . ‘Hs’ designates that Scott is a featherless biped.

The difference between these two statements is analogous to that be-
tween 38-1 and 38-2: Statement 38-6, though not itself a rule, follows

173 IV. ON METALANGUAGES FOR SEMANTICS

from the semantical rules alone, while for the derivation of 38-7 a factual
premise is needed.

The foregoing discussion shows that, even if somebody possesses no
other information concerning S, and Sj than the semantical rules for these
systems formulated in he is, nevertheless, in a position to know the
meanings — that is to say, not only the extensions but also the intensions —
which are intended, first, for the primitive descriptive constants and,
second, for all designators. All he has to do is to look, first, at the rules of
designation themselves and, second, at those statements about designa-
tion which follow from the semantical rules alone, leaving aside all those
statements in M, which, although true, can be arrived at only with the
help of factual knowledge. In other words, he has to consider only those
statements about designation which are L-true in M,.

It is sometimes said that a metalanguage, in which the semantics of an
object language S is to be formulated, must contain translations of all
expressions or at least of all designators in 5 . If this were right, would
not suffice as a semantics language for S„ because cannot, of course,
contain an expression L-equivalent to the intensional sign ‘N’ in S,. But
the requirement mentioned is only approximately right; strictly speaking,
it is too strong. The metalanguage must, indeed, contain for every sen-
tence in 5 an L-equivalent sentence; furthermore, it must be sufficiently
equipped with variables and descriptive expressions. It is, however, not
necessary that it contain an L-equivalent expression for every logical sign
in S. Although M, cannot contain a translation of ‘N’, it can contain a
semantical rule for ‘ N’, for instance, the rule of ranges mentioned ahove.
If ©i is a sentence in Sj containing ‘N’, then an extensional language like
S, or Me cannot, of course, contain a translation of ©, in the strong sense
of a sentence with the same intensional structure (§ 14). But it can be
shown that S,, and hence Me, too, always contains a sentence L-equivalent
to ©,. [For full sentences of ‘N’, this follows simply from the circumstance
that they are either L-true or L-false (see 39-2) ; however, since sentences
may contain several occurrences of ‘N’ and quantifiers in any combina-
tion, the general proof is rather complicated.] Further, S, and Sj contain
the same variables and descriptive signs. Hence, if M^ is sufficient for the
formulation of the semantics of Si, it is likewise sufficient for that of S,.

On the basis of these considerations, I am inclined to believe that it is
possible to give a complete semantical description even of an intensional
language system like S3 in an extensional metalanguage like M,. However,
this problem requires further investigation.

CHAPTER V

ON THE LOGIC OF MODALITIES

In this chapter we study logical modalities like necessity, possibility, im-
possibility. We introduce ‘N’ as a symbol of necessity; the other modal con-
cepts, including necessary implication and necessary equivalence, can be de-
fined with its help. The modal system Si is constructed by adding ‘N’ to our
previous system S, (§ 39); and the semantical rules for Sa are stated (§41).
An analysis of the variables occurring in modal sentences shows that they have
to be interpreted as referring to intensions (§ 40) ; hence a translation in words
must be given either in terms of intensions (in the metalanguage M) or in
neutral terms (in M') (§ 43). Quine’s views on the possibility of combining
modalities and variables are discussed (§44). Finally, the main results of the
discussions in this book are briefly summarized (§ 4S).

§ 39 . Logical Modalities

We form the modal system Sa from our earlier system S, by the addition
of the modal sign ‘N’ for logical necessity. We regard a proposition as necessary
if any sentence expressing it is L-true. Other modalities can be defined in terms
of necessity, for example, impossibility, possibility, contingency. With the help
of ‘N’, we define symbols for necessary implication and necessary equivalence;
the latter symbol may be regarded as an identity sign for intensions.

In the earlier chapters, modal sentences have sometimes been taken
as examples, especially sentences about necessity or possibility, either in
words (for instance, in §§ 30 and 31) or in symbols (for instance, § ii,
Example II). We use ‘N’ as a sign for logical necessity; ‘N(A)’ is the sym-
bolic notation for ‘it is (logically) necessary that A’.

Quite a number of different systems of modal logic have been con-
structed, by C. I. Lewis (see Bibliography) and others.' These systems
differ from one another in their basic assumptions concerning modalities.
There is, for instance, the question of whether all sentences of the form
‘N^ 3 NN/>’ are true, in words: ‘if it is necessary that p, then it is neces-
sary that it is necessary that p\ Some of the systems give an affirmative
answer to this question, other systems give a negative answer or leave it
undecided. Not only do logicians disagree among themselves on this ques-
tion, but sometimes also one logician constructs systems which differ in
this point, probably because he is doubtful whether he should regard the
sentences mentioned as true or false. There are several further points of

' For bibliographical references up to 1938, see Church’s bibliography in Journal of Symbolic
Logic, Vols. I and III; the pertinent references are listed in III, 199 (“Modality”) and 202
(“Strict Implication”).

173

174

V. ON THE LOGIC OF MODALITIES

difference between the systems. All these differences are, I think, due to
the fact that the concept of logical necessity is not sufficiently clear; it can,
for instance, be conceived in such a way that the sentences mentioned
are true, but also in another way such that they, or some of them, are false.

Our task will be to find clear and exact concepts to replace the vague
concepts of the modalities as used in common language and in traditional
logic. In other words, we are looking for explicata for the modalities. It
seems to me that a simple and convenient way of explication consists in
basing the modalities on the semantical L-concepts. The concept of logical
necessity, as explicandum, seems to be commonly understood in such a
way that it applies to a proposition p if and only if the truth of p is based
on purely logical reasons and is not dependent upon the contingency of
facts; in other words, if the assumption of not-/> would lead to a logical
contradiction, independent of facts. Thus we see a close similarity between
two explicanda, the logical necessity of a proposition and the logical
truth of a sentence. Now for the latter concept we possess an exact ex-
plicatum in the semantical concept of L-truth, defined on the basis of the
concepts of state-description and range {2-2). Therefore, the most natural
way seems to me to take as the explicatum for logical necessity that
property of propositions which corresponds to the L-truth of sentences.
Accordingly, we lay down the following convention for ‘N’ :

39 - 1 . P'or any sentence ‘. . .’, ‘N(. . is true if and only if . .’ is L-true.
¥

We shall construct the system Sj by adding to the system S, the sign ‘N’
with suitable rules such that the convention just stated is fulfilled (§ 41).
This convention may be regarded as a rule of truth for the full sentences
of ‘N’. Sj thus contains all the signs and the sentences of S,.

On the basis of our interpretation of ‘N’, as given by the convention
39-1, the old controversies can be solved. Suppose that ‘L-true in S^’ is
defined in such a way that our earlier convention 2-1, which says that a
sentence is L-true if and only if it is true in virtue of the semantical rules
alone, independently of any extra-linguistic facts, is fulfilled. Let ‘A’ be
an abbreviation for an L-true sentence in Sj (for example, ‘Hs V ~ Hs’).
Then ‘N(A)’ is true, according to 39-1. And, moreover, it is L-true, be-
cause its truth is established by the semantical rules which determine the
truth and thereby the L-truth of ‘A’, together with the semantical rule for
‘N’, say 39-1. Thus, generally, if ‘N(. . .)’ is true, then ‘NN(. . .)’ is true;
hence any sentence of the form D NN/>’ is true. This constitutes an
affirmative answer to the controversial question mentioned in the be-
ginning. It can be shown in a similar way that every sentence of the

§ ». LOGICAL MODALITIES 17S

form D N ^ N^’ is true. This settles another one of the contro-

versial questions.'

T his analysis leads to the result that, if ‘N(. . .)’ is true, it is Lr-true;
and if it is false, it is L-false; hence:

39-2. Every sentence of the form ‘N(. . .)’ is L-determinate.

Therefore, the convention 39-1 may be replaced by the following more
specific one :

39-3. For any sentence '. . .’ in Sj, ‘N(. . .)’ is L-true if ‘. . is L-true;
and otherwise ‘N(. . is L-false.

On the basis of the concept of logical necessity, the other logical
modalities can easily be defined, as is well known. For example, ‘p is im-
possible’ means ‘non-/» is necessary’; ‘p is contingent’ means ‘p is neither
necessary nor impossible’ ; ‘p is possible’ means ‘p is not impossible’ (we
adopt this interpretation in agreement with the majority of contemporary
logicians, in distinction to other philosophers who use ‘possible’ in the
sense of our ‘contingent’). Let us use the diamond, ‘0 ’, as a sign of possi-
bility ; we define it on the basis of ‘N’ :

39-4. Abbreviation. ‘<}(- • •)’ for ‘~N ~(. . .)’.

It would also be possible to take ‘0 ’ as primitive, as Lewis does, and then
to define ‘N(. . .)’ by ‘~ () ~(. . .)’.

There are six modalities, that is, purely modal properties of proposi-
tions (as distinguished from mixed modal properties, for instance, con-
tingent truth, see 30-1). The accompanying table shows how they can be

THE SIX MODALITIES

Modal Property of
a Proposition

With ‘N‘

With • 0’

Semantical Prop-
erty of a Sentence

Necessary .

Np

0 /“V/ p

L-true

Impossible, . . .

N~p

~ 0 p

L-false

Contingent . ,

N p

0 ~ ^ • Op

Factual

Non-necessary

~N/.

0 ~ p

Non-L-true

Possible

^ N ^

Op

Non-L-false

Noncontingent. . j

N^ V N ~ ^

py 0 p j

L-determinate

expressed in terms of ‘N’ and in terms of ‘O’- The last column gives the
corresponding semantical concepts; a proposition has one of the modal

^ The two questions and the reasons for our affirmative answers are discussed in more detail
in [Modalities], $ 1 .

176 V. ON THE LOGIC OF MODALITIES

properties if and only if any sentence expressing the proposition has the
corresponding semantical property.

Every proposition with respect to a given system 5 is either necessary
or impossible or contingent. This classification is, according to our inter-
pretation of the modalities, analogous to the classification of the sentences
of S into the three classes of L-true, L-false, and factual sentences. There
is, however, one important difference between the two classifications. The
number of L-true sentences may be infinite, and it is, indeed, infinite
for each of the systems discussed in this book. On the other hand, there is
only one necessary proposition, because all L-true sentences are L-equiva-
lent with one another and hence have the same intension. [This result
holds only for that use of the term ‘proposition’ which is based on L-
equivalence as the condition of identity. It is, of course, possible to choose
a stronger requirement for identity, for instance, intensional isomorphism.
In this case the intensional structures are called ‘propositions’. And their
number is infinite.] Likewise, there is only one impossible proposition, be-
cause all L-false sentences are L-equivalent. But the number of con-
tingent propositions (with respect to a system with an infinite number of
individuals) is infinite, like that of factual sentences.

It should be noted that the two sentences ‘N(A)’ and ‘the sentence ‘A’
is L-true in S 2 ’ correspond to each other merely in the sense that, if one
of them is true, the other must also be true; in other words, they are L-
equivalent (assuming that L-terms are defined in a suitable way so as
to apply also to the metalanguage). This correspondence cannot be used
as a definition for ‘N’, because the second sentence belongs, not to the
object language as the first one does, but to the metalanguage M. The
second sentence is not even a translation of the first in the strict sense
which requires not only L-equivalence but intensional isomorphism
(§ 14). If M contains the modal term ‘necessary’, then ‘N(A)’ can be
translated into M by a sentence of the form ‘it is necessary that . . .’
(where ‘. . .’ is the translation of ‘A’). If M contains no modal terms,
then there is no strict translation for ‘N(A)’. But the correspondence
stated makes it possible in any case to give an interpretation for ‘N(A)’ in
M with the help of the concept of L-truth, for example, by laying down
the truth-rule, 39-1.

On the basis of ‘N’, we introduce two further modal signs for modal
relations between propositions:

39-6. Abbreviation. Let ‘. . .’ and ‘ ’ be sentences in S^. ‘. . . 5 ’

for ‘N(. . . D ---)’.

§ 40. MODALITIES AND VARIABLES

177

39 - 6 . AbbreviaHon. Let and ‘ ’be any designators in S, (sentences

or otherwise). ‘. . . s ’for ‘N(. . . = )’.

Thus ‘ 5 ’ is a sign for necessary implication between propositions (Lewis’
strict implication). The symbol ' = ’ is a sign for necessary equivalence.
The sign ‘ = ’ in Sa is the analogue to the term ‘L-equivalent’ in its non-
semantical use in M (5-4) or M' (§ 34), where it designates a relation be-
tween intensions, not between designators. When standing between
sentences, it corresponds to Lewis’ sign ' = ’ for strict equivalence. We
have seen earlier that ‘ = ’, standing between designators of any type, is a
sign for the identity of extensions (see remark on 5-3). Here in S,, ‘s’ is,
similarly, a sign for the identity of intensions. For example, ‘H s RA’ is
short for ‘N(H s RA)’. Hence, according to the rule 39-r, ‘H s RA’ is
true if and only if ‘H s RA’ is L-true, hence if and only if ‘H’ and ‘RA’
are L-equivalent, in other words, have the same intension.

We have earlier formulated the two principles of interchangeability
(i2-i and 12-2). For the first principle we have given, in addition to the
chief formulation in semantical terms (12-ia), alternative formulations
with the help of sentences of the object language containing ‘ = ’ (12-ib
and c) . Now, with the help of ‘ ^ we can provide analogous formulations
for the second prmciple. The following theorems 39-7b and c, which may
be added to i2-2a as i2-2b and c, follow from i2-2a because SI,- and
are L-equivalent if and only if 31 , • s 81 * is true.

Second Principle of Interchangeability (alternative formulations) :

39 - 7 . Under the conditions of 12-2, the following holds:

b. (i2-2b). ( 3 Iy ^ 31 *) D (. . 3 l,- . . = . . 81 * . .) is true (in S).

c. (i2-2c). Suppose the system 5 contains variables for which 31 , •
and 31 * are substitutable, say ‘«’ and 'v'-, then ‘(«)(ii) [(« m v) D
(. . M . . = . . j) . .) ] ’ is true (in S) . -

§ 40 . Modalities and Variables

Problems concerning the interpretation of variables in modal sentences are '
discussed, in preparation for the semantical rules given in the next section. It is
found that a universal quantifier preceding ‘N’ is to be interpreted as if it fol-
lowed the ‘N’. It is generally shown that variables in modal sentences are to be
understood as referring to intensions rather than to extensions. Thus an indi-
vidual variable in S, is interpreted as referring to individual concepts rather
than to individuals. We decide to take as values of these variables not only
those individual concepts which are expressible by descriptions in S, but the
wider class of all individual concepts with respect to S,. A concept of this kind
is represented by any assignment of exactly one individual constant to each
state-description in S,.

178

V. ON THE LOGIC OF MODALITIES

So far we have given an interpretation for 'N’ only in the case in which
the argument-expression of ‘N’ is a sentence. But in a system which con-
tains variables we also have to solve the problem of interpreting occur-
rences of ‘N’ followed by a matrix with free variables, e.g., ‘N(Pr)’. Let
us investigate this problem in a general way for a system S containing a
variable of any type. How should we interpret the sentence ‘(“)[N
(. . M . .)]’, where is a matrix containing ‘u’ as the only free vari-

able? Let us first consider the case in which ‘u’ has only a finite number of
values, say n, and all these values are expressible in S, say by the designa-
tors ‘U,’, ‘Uj’, . . . ‘U„’. (As we shall see later, the interpretation of a
variable in a modal sentence has to be given in terms of value-intensions,
not value-extensions. Therefore, the statement just made is to be under-
stood as saying that there are n value-intensions for ‘u’ and that they are
the intensions of the designators ‘U,’, etc.) Now any universal sentence,
whether in an extensional or in a modal language, always means that all
values of the variable possess the property expressed by the matrix.
Therefore, if the number of values is n, the universal sentence means the
same as the conjunction of the n substitution instances of the matrix.
In our example, ‘(m)[N(. . u . .)]’ means the same as ‘N(. . U, . .) •
N(. . U, N(. . U„ . .)’.

A conjunction of n components (» ^ 2) is L-true if and only if every one
of the components is L-true. Therefore, the following holds, in virtue of
the correspondence between necessity and L-truth (39-1):

40 - 1 . If ‘A,’, . . . ‘An’ are any sentences, ‘N(Aj • Aj • . . . • A„) ’ is L-
equivalent to ‘N(A.) • N(A3) • . . . • N(A„)’.

If we apply this to the above result, we find that ‘(«)[N(. . u . .)]’ means
the same as ‘N[(. . Ui ..)•(.. U, U„ . .)]’ and hence

the same as ‘N[(m)(. . m . .)]’. Thus the result is that ‘{u)’ and ‘N’ may
exchange their places.

Next, let us consider the case in which the variable has an infinite,
but denumerable, number of values, all of which are expressible in S, say
by the designators ‘U,’, ‘Uj’, etc. Here we cannot form a conjunction of
the substitution instances, but we can still consider their class. If we
interpret a class of sentences as a joint assertion of its sentences, in accord
with the usual procedure, then we can apply semantical concepts to it in
the following way : We define the range of a class of sentences as the prod-
uct of the ranges of the sentences. This leads to the following two re-
sults:

§ 40. MODALITIES AND VARIABLES

179

(i) A class of sentences is true if and only if all its sentences are true.

(ii) A class of sentences is L-true if and only if all its sentences are
L-true.

Now the sentence ‘(«)[N(. . u . .)]’ is true if and only if the class of the
instances ‘N(. . U„ . .)’ for » = i, 2, etc., is true; hence, according to (i),
if and only if every sentence of the form ‘N(. . U„ . .)’ is true; hence, ac-
cording to 39-1, if and only if every sentence of the form ‘. . U„ . .’ is
L-true ; hence, according to (ii), if and only if the class of these sentences is
L-true; hence, if and only if '(«)(. is L-true; hence, according to

39-1, if and only if ‘N[(m)(. . u . .)]’ is true. Thus the result is that, in
the case of infinitely many values also, the quantifier '(«)’ and the modal
sign ‘N’ in the original sentence may exchange places.

It seems natural to apply the same result to the case in which not all
values of ‘u’ are expressible in S, that is to say, to interpret a sentence of
the form ‘(m)[N(. . u . .)]’ in any case, irrespective of the number and ex-
pressibility of the values of ‘u\ as meaning the same as ‘N [(«)(.. m . .)]’.
In particular, we shall construct the semantical rules of the system S, in
such a way that any two sentences of the forms just stated are L-equiva-
lent (§ 41). In Sj ‘u’ must, of course, he an individual variable.

Since a modal system contains not only extensional but also intensional
contexts, a designator may, in general, be replaced by another one only if
they are not merely equivalent but L-equivalent. Thus, in general, we
have to take into consideration the intensions of the designators, not
merely their extensions. Similarly, we have to consider for a given vari-
able its value-intensions in the first place. If the system contains variables
of the type of sentences, say ‘p’, etc., then a quantifier with a variable
of this kind occurring in a modal sentence must be interpreted as re-
ferring to propositions, not to truth-values. For example, the sentence
‘( 3 />) (~N/>) ’ must be understood as saying that there is a non-necessary
proposition. It would hardly make sense to interpret it as saying that
there is a non-necessary truth-value, because there are propositions
with the same truth-value such that one of them fulfils the matrix
while another one does not. This interpretation in terms of
propositions seems generally accepted. C. I. Lewis, as well as the other
logicians who have discussed his systems of modal logic or have con-
structed new ones, have used interpretations in terms of propositions. If
variables of the type of predicators of degree one occur in a modal system,
it is clear that they must be interpreted analogously in terms of proper-
ties, not of classes. Here, again, I think that most logicians would agree;

i8o V. ON THE LOGIC OF MODALITIES

however, modal sentences with v^lriables of this kind have not been dis-
cussed frequently.

In my view the situation with respect to individual variables is quite
analogous, although this is usually not recognized. I think that individual
variables in modal sentences, for example, in Sa, must be interpreted as
referring, not to individuals, but to individual concepts. The difficulties
which would otherwise arise will be explained later (§43). Thus a sentence
of the form ‘(x)(. in Sj is to be interpreted as referring to all

individual concepts. Therefore, we now have to study the question as to
what is to be regarded as the totality of all individual concepts with re-
spect to Sj.

We shall assume for the following discussions that the individual con-
stants in Sa are L-determinate (§ 19), that is to say that they are inter-
preted by the rules of dosignation as referring to positions in an ordered
domain and that any two different constants refer to different positions.
[For this purpose, it would be more natural to construct Sa on the basis of
Sj (§ 18) rather than of S,. The reason for taking Si as the basis is merely
the possibility of using the earlier examples. But we must then suppose
that, for example, the rule of designation for ‘s’ does not use the phrase
‘the man who was known by the name of ‘Walter Scott’ ’, but rather:
‘the man who was born at such and such a place at such and such a
time’; and even this formulation would not be entirely adequate.] Conse-
quently, we take any sentence of the form ‘a = b’ as L-false. However,
= -sentences with one or two descriptions (for example, ‘{rx) (Axw) = s’)
are still, in general, factual.

A description 31 , in Sj, say ‘{ix} (. . x . characterizes one of the indi-
vidual positions with the help of the property expressed by the matrix
\ . X . If exactly one position has this property, then this position
is the descriptum; otherwise, a* is the descriptum (§ 8). Thus for the
determination of the descriptum, the extension of 2l„ factual investiga-
tion is required (unless the description is L-determinate). On the other
hand, the intension of 31 ,, the individual concept expressed by 3 l„ must be
something that can be determined by logical analysis alone. In order to
understand more clearly what kind of entity an individual concept is,
let us see what we can find out about the description 3 l< by logical analysis
alone. Suppose a state-description in Si is given (which is an infinite
class of sentences in Si). Then the question of whether or not there is
exactly one individual position in fulfilling the matrix ‘. . x . .’ — in
other words, whether or not there is exactly one substitution instance of

§ 40. MODALITIES AND VARIABLES i8i

the matrix with an individual constant which holds in — is a purely
logical question. If the answer is in the affirmative, the descriptum of 8,-
with respect to is represented by that one individual constant; other-
wise it is represented by 'a*’. Thus the description 2[, assigns to every
state-description exactly one individual constant; any individual con-
stant may be assigned to several state-descriptions. If 2li and SI/ are L-
equivalent and hence express the same individual concept, then both
assign to any state-description the same individual constant. Therefore,
we might say that an individual concept with respect to Sa is an assign-
ment of exactly one individual to every state (which is a proposition ex-
pressed by a state-description). However, we shall actually take not these
states but the state-descriptions; and not the individuals but the indi-
vidual constants. The latter is possible because we have assumed that
these constants are L-determinate and that there is a one-one correlation
between the individuals and the individual constants. Thus we shall take
any assignment of exactly one individual constant to each state-descrip-
tion in Sj (in other words, any function from state-descriptions to indi-
vidual constants) as representing an individual concept with respect to S,.
Only a small part (a denumerable class) of the individual concepts repre-
sented by assignments of this kind are expressible by descriptions in Sa.
Now we decide to take as values of the individual variables in Sa not only
the individual concepts expressible by descriptions in Sa but all individual
concepts represented by assignments of the kind described; we call them
individual concepts with respect to Sa. In the next section we shall lay down
the semantical rules for S, in accord with this decision; a universal quanti-
fier will be interpreted as referring to all individual concepts with respect
to Sa.

Some remarks may, incidentally, be made concerning the interpreta-
tion of variables of other than individual type. Let 5 be a modal system
which also contains propositional variables etc., and variables etc.,
for properties of level one, that is, properties of individuals. As values for
propositional variables we should take not only those propositions which
are expressed by sentences in S, but all propositions with respect to S.
They are represented by the ranges in S, that is, the classes of state-
descriptions in 5. And as values for ‘/’, etc., we should take not only those
properties which are expressed by predicators (including lambda-expres-
sions) in 5, but all properties with respect to S. Since the attribution of a
property to an individual results in a proposition, we may regard a

V. ON THE LOGIC OF MODALITIES

182

property as an assignment of exactly one proposition to each individual.
Therefore, we may represent the properties with respect to S by the as-
signments of ranges (classes of state-descriptions) in 5 to the individual
constants in S. Similarly, assignments of ranges in S to ordered pairs of
individual constants in 5 may be taken as representing the relations with
respect to S as values of relation variables in 5 . [In analogy to the rules of
ranges for matrices containing individual variables in S,, which will be
given in the next section, rules for variables of other types in S might be
stated as follows: (i) The matrix '/>’ holds in the state-description ifn for
a certain range as value if and only if belongs to this range, (ii) The
matrix ‘/a’ holds in for a given alignment of the kind described as
value of ‘/’ if and only if S:„ belongs to that range which is assigned to
‘a’.]

§ 41 . Semantical Rules for the Modal System S,

On the basis of our previous decisions concerning the interpretation of ‘N’
(§ 39) and of the individual variables in Sa (§ 40), we lay down semantical rules
for Sa. The most important rules are the rules of ranges, which are here some-
what more complicated than for S, because individual concepts rather than indi-
viduals must here be taken as values of the variables. The L-concepts for Sa
have the same definitions as for S,. Some examples of L-true modal sentences in
Sa are given.

The signs of the modal system S, comprise those of St and, in addition,
the modal sign ‘N’. In S,, compound designators and designator matrices
are formed out of atomic matrices with the help of the following means:
the ordinary (i.e., nonmodal) connectives, quantifiers, the iota-operator,
and the lambda-operator. In Sa a rule of formation for ‘N’ is added,
which says that, if ‘. . .’is any matrix, ‘N(. . .)’ is a matrix.

Now we have to construct the rules of ranges for Sa. The state-descrip-
tions in Sa are the same as in S, (§ 2), because Sa does not contain any
new descriptive constants. If we had only sentences without variables, we
could simply take the rules of ranges for S, (see the examples in § 2,
omitting the rule for a universal sentence) and add the following rule :

41 - 1 . N(®<) holds in every state-description if ©, holds in every state-
description; otherwise, N(@,) holds in no state-description.

This rule is clearly in accord with our convention 39-3 (see 2-2 and 2-4).
However, in order to accommodate sentences with variables, we have to
use, instead, more complicated rules of ranges. They must apply not only
to sentences, like the rules of ranges for S, (§ 2), but to matrices, and they

1 41. SEMANTICAL RULES FOR THE MODAL SYSTEM S, 183

must refer to values of the individual variables occurring in the matrix.
According to our analysis in the preceding section, we take as values of
the variables all individual concepts with respect to S,; every one of these
concepts is represented by an assignment of individual constants to state-
descriptions. Suppose that we have chosen as a value of the variable ‘x’
occurring in the atomic matrix T*’ an assignment of this kind and that
the individual constant assigned to a given state-description is ‘b’.
Then the question of whether the matrix T*’ for the chosen value of ‘x’
holds in means simply whether the sentence Tb’ holds in and this
is, of course, the case if ‘Pb’ belongs to (compare the example (i)
of the rules of ranges for S, in § 2). This analysis suggests the first of the
subsequent rules of ranges (4i-2a). The other rules are analogous to the
rules of ranges for Sx (§ 2), together with the rule 41-1 for ‘N’, except that
the present rules apply to matrices and therefore have to refer to assign-
ments as values of the free variables.^ Note that sentences are matrices
without free variables (§ 1) ; therefore, these rules apply also to sentences,
in which case the references to values are dropped.

41-2. Rules of ranges for the modal system Sj. Let 8l< be a matrix and
Sn be a state-description in Sj. By a value of a variable we mean
any assignment of the kind described earlier.

a. Let SI. be of atomic form. Sl< holds in for given values of the
individual variables occurring in Sl<, if and only if contains
the atomic sentence formed from 81, by substituting for every
free variable the constant assigned to ffn by the value of the
variable.

b. Let 31, be an = -matrix with individual signs (constants or vari-
ables) . 81, holds in „ for given values of the variables occurring
in 3l„ if the individual constant for the left side (that is, either

> The system MFL described in [Modalities], § 9, is similar to, but somewhat simpler than,
our present system S,. Sentences of the form ‘a = b’ in MFL are regarded as L-false, like the
corresponding sentences of the form ‘a s b’ in S,; this shows that the individual constants in
MFL are, in terms of our present theory, L-determinate like those in S.. The state-descriptions
are the same in both systems. The differences are as follows; MFL does not contain lambda-
expressions and individual descriptions; this difference is not essential, since both kinds of
expressions in S, can be eliminated, as we have seen. More essential is the difference in the
interpretation of individual variables. A universal sentence ‘(*)(. in MFL is regarded

as L-equivalent to the class of substitution instances of the matrix with all individual

constants; thus, in terms of our present theory, the universal quantifier refers to all L-dctermi-
nate individual concepts and to no others. A universal quantifier in S., on the other hand, refers
to all individual concepts (with respect to Sj). This wider range of values for the individual
variables in S, seems more adequate; but it makes necessary the somewhat more complicated
form of the rules of ranges as given in the text, while the rules of ranges for MFL are as simple
as those for S,, together with the rule 41-1 for ‘N’.

184 V. ON THE LOGIC OF MODALITIES

the individual constant standing on the left side or the indi-
vidual constant assigned to by the value of the variable
standing on the left side) is the same as that for the right side.

c. Let Sli be ~Slj. ?li holds in for given values of the variables
occurring freely in 2l<, if Sly does not hold in for these values.

d. Let be 21, V SI*, a, holds in for given values of the free
variables, if either Sly or 81 * or both hold in for these values.

e. Let SI, be Sly • SI*. SI, holds in for given values of the free
variables, if both Sl< and 8* hold in for these values.

f. Let SI, consist of a universal quantifier followed by the matrix 8,
as its scope. Sly holds in i?* for given values of the variables
occurring freely in Sly (hence not including the variable occurring
in the initial quantifier), if Sly holds in for every value of the
variable of the initial quantifier and the given values of the
other free variables.

g. Let Sly be N(8ly). Sly holds in for given values of the free vari-
ables, if Sly holds in every state-description for these values.

The following two theorems are simple consequences of these rules;
they may be used instead of the rules for the determination of the range
of a nonmodal matrix or sentence in S,.

41 - 3 . Let Sly be a matrix of any form without ‘N’ in Sj. 81 , holds in for
given values of the free variables, if and only if the sentence formed from
8[y by substituting for every free variable the constant assigned to S'n by
the value of the variable holds in

41 - 4 . If a sentence in S, does not contain ‘N’, then it holds in Sj in the
same state-descriptions as in S,.

In order to avoid certain complications, which cannot be explained
here, it seems advisable to admit in only descriptions which do not con-
tain ‘N’. But any description may, of course, occur within the scope of an
‘N’. The smallest matrix in which a description occurs (in the primitive
notation) is always a nonmodal context, because the description must be
an argument expression either of a primitive predicator constant or of
‘s’. This smallest matrix is then taken as the context ‘- -(ix) (. . x . .) -
which can be transformed into 8-2. In this way every description can be
eliminated. Since L-equivalent sentences are L-interchangeable also
within modal contexts, according to the second principle of interchange-
ability (12-2), the result of the elimination is L-equivalent to the original
sentence; or, rather, we lay down a rule to the effect that any sentence con-

§ 41 . SEMANTICAL RULES FOR THE MODAL SYSTEM Sj 185

taining descriptions holds in the same state-descriptions as the sentence
resulting from the described elimination of the descriptions, and hence the
two sentences become L-equivalent.

Another point is worth noting. Although we interpret the individual
variables in Sj as referring to individual concepts, not to individuals,
nevertheless a description in S, characterizes, not one individual concept,
but mutually equivalent individual concepts — in other words, one indi-
vidual. This follows from the rule just mentioned, which permits the trans-
formation into 8-2. The first part of 8-2 says, in words; ‘there is an indi-
vidual concept y such that, for every individual concept x, x has the de-
scriptional property if and only if * is equivalent (not ‘L-equivalent’ or
‘identical’!) to y’; in other words, ‘all individual concepts equivalent to y,
and only these, have the property’; or, ‘the individual y is the only indi-
vidual which has the property’. This is as it should be, because the purpose
of a description, even in a modal language, is to refer to one individual
with the help of a property possessed by that individual alone. Neverthe-
less, the description has, of course, a unique intension, which is an indi-
vidual concept. This individual concept is not the only one possessing
the descrip tional property, since, as we have seen, all equivalent ones do
likewise ; but it is uniquely determined by the descriptional property ; as
Frege puts it, it is not the individual but the way in which the description
refers to the individual.

For lambda-expressions we do not impose the restriction stated for
descriptions; they may also contain ‘N’. Any lambda-operator can be
eliminated in S3 by conversion in the same way as in S, (§ i). Here, again,
a rule would be laid down saying that a sentence containing lambda-
operators holds in the same state-descriptions as the sentence resulting
from their elimination.

The L-concepts are defined for Sj in the same way as for S, (§ 2), The
following theorems give a few results, which hold on the basis of the rules
of ranges stated above.

41 - 6 . Any sentence of one of the following forms is L-true in Sj. (The
variables ‘p’, ‘q\ . . ‘/’, do not occur in S3 but are here used merely
to describe forms of sentences in S3. A sentence in S3 is said to have
one of the forms described if it is formed by substituting for ‘/>’ or
‘9’ any sentence in Sa and for ‘/ar’ any matrix containing ‘x' as the
only free variable.)

t86

V. ON THE LOGIC OF MODALITIES

a. ‘N/> 3 p’.

b. ‘p 3 <)p’.

c. ‘(p 3 9 ) 3 (N/» 3 NqY

d. ‘N{p 0 q) s Np u Nq’.

e. ‘0iP^9) s <>/> V Og.

f. ‘NN/> ^ Np\

g. ‘N ~ N/> s ~ N/>’.

h. '00/> - 0/>’-

i. ‘ON^ ^ Np’.

j. 'N 0 /> ^ Op'-

k. ‘(x)N(/x) m N(x)(/x)\

l . Y^x)N(/x) 3 N(aa:)(/a;)’.

m. ‘{'3.x)0(fx) =

n- '0 (*)(/*) ^ (*)0(/*)’-

We see from these theorems that ‘N’ is quite similar to a universal
quantifier and to an existential quantifier. This seems plausible, since
N©,- is true if ©< holds in every state-description, and 0 ®» is true if ©<
holds in at least one state-description.

§ 42. Modalities in the Word Language

The problem of the translation of modal sentences of S2 into the metalan-
guages M and M' is discussed. It is shown that it is advisable to use for the
translations either terms of intensions in M or neutral terms in M'. The use of
terms of extensions within modal sentences in M is not in itself incorrect, pro-
vided that certain restrictions are observed; but it involves the danger of mak-
ing wrong inferences by overlooking the restrictions.

We shall examine here the problem of the formulation of modal sen-
tences in words and, in particular, the problem of the translation of
modal sentences into our metalanguages M and M'. It is worth while to
study this problem because, it seems to me, certain difficulties which have
sometimes been found in connection with modal sentences are due chiefly
to their inadequate or misleading formulation in the word language.

Since modal sentences, for instance, in S, or in a richer language with
several types of variables, are not semantical, their translations are like-
wise not semantical sentences and hence belong to the nonsemantical part
of M and M' (this part of M' was explained in §§ 34-36). As translation
of ‘N’, we take ‘it is necessary that’; hence, this is an intensional phrase.

We shall discuss three examples — A, B, and C. In A, we have predica-
tors as argument expressions of ' = ’ or ‘s’; in B, sentences; in C, indi-
vidual expressions. Otherwise, the three examples are perfectly analogous.
Therefore, we arrange them in three parallel columns. This facilitates the

§ 42. MODALITIES IN THE WORD LANGUAGE i8r

comparison of corresponding expressions in the three exan^les and the
recognition of their analogy.

Because of the perfect analogy, any one of the three examples would
theoretically be sufficient. However, for practical reasons it seems advis-
able to give all three. The purpose of the analysis of the examples is to
show that it is advisable to formulate modal sentences either in terms of
intensions or in neutral terms, while formulation in terms of extensions
involves certain dangers. Now this result is easily seen in the case of
predicators; presumably, most readers will agree in this case. Then the
analogy will make it easier to recognize the same situation in the case of
sentences and, finally, in the case of individual expressions. In this last
case the inhibitions against a translation in terms of intensions are strong-
est because it is not customary to speak of individual concepts. Therefore,
here the help of the two other examples seems necessary for practical,
psychological reasons, although theoretically the situation is here as clear
and simple as in the first two cases.

The example A (the conjunction of 4?-iA and 42-2aA) is similar to one
given by Church;"' our ‘'~N(. . .)’ corresponds to his ‘ 0 ~(- • •)’• I® tlie
example C, we use ‘au’ as abbreviation for ‘(iz) (Artv)’. In the translation
of this description into the word language, we omit, for the sake of brevity,
the phrase ‘or a*, if there is not exactly one such individual’ (as we did
earlier, § 9).

The following sentences in Sj are true but not L-true (see 3-7 and 9-2) :
42-1. A B C

‘F,B=H’. ‘(F.B)s = Hs’. ‘au = s’.

Therefore, according to 39-1, prefixing ‘N’ yields false sentences; hence
the following is true:

42-2a. A B C

‘~N(F,B = H)’; ‘~N[(F.B)s = Hs]’; ‘~N(au = s)’;

or, abbreviated with ‘ = ’ (39-6) :

42-2b. A B C

‘~(F,BsH)’. ‘~[(F,B)s ^ Hs]’. ‘~(au s s)’.

Now let us examine the question of the translations of these sentences
of S, into M. The first sentence, 42-1 (in each of the three examples), is a
nonmodal sentence. It can be translated in two different ways, either into
42-3 in terms of intensions with the nonsemantical term ‘equivalent’ (see
5-3 and 5-5) or into 42-4 in terms of extensions with the identity phrase
‘is the same as’ (see 4-7 and 9-1) :

< [Review Q.J, p. 46.

i88

V. ON THE LOGIC OF MODALITIES

42-3. A

‘The property Feath-
erless Biped is equiv-
alent to the property
Human’.

B

‘The proposition
that Scott is a
featherless biped is
equivalent to the
proposition that
Scott is human’.

C

‘The individual
concept The Au-
thor Of Waverley
is equivalent to the
individual concept
Walter Scott’.

42-4. A

‘The class Feather-
less Biped is the
same as the class
Human’.

B

‘The truth-value
that Scott is a
featherless biped is
the same as the
truth- value that
Scott is human’.

C

‘The individual The
Author Of Waver-
ley is the same as
the individual
Walter Scott’.

For the modal sentences 42-2, however, the situation is different. First,
we shall give the translation into M in terms of intensions. We base the
translation 42-5 on the second of the two notations a and b given for 42-2,
utilizing the fact that ‘^’ is a sign for the identity of intensions (§ 39).
(For A, see 4-8; for B, 6-4; for C, § 9).

42-6. ABC

‘The property
Featherless Biped is
not the same as the
property Human’.

‘The proposition that
Scott is a feather-
less biped is not the
same as the propo-
sition that Scott is
human’.

‘The individual
concept The Au-
thor Of Waverley
is not the same as
the individual con-
cept Walter Scott’.

This translation is adequate and unobjectionable. Not so, however, the
following translation in terms of extensions; here we base the translation
on the first notation 42-2a and regard ‘ = ’ as a sign for the identity of
extensions (see remark on 5-3) .

42-6. A

‘It is not necessary
that the class Feath-
erless Biped is the
same as the class Hu-
man’.

B

‘It is not necessary
that the truth-value
that Scott is a
featherless biped is
the same as the
truth-value that
Scott is human’.

C

‘It is not necessary
that the individual
The Author Of
Waverley is the
same as the individ-
ual Walter Scott’.

§ 42. MODALITIES IN THE WORD LANGUAGE

189

Formulations of this kind might perhaps be admitted as sentences in M ; if
so, they would presumably be regarded as true and as correct translations
of 42-2a. However, these formulations are dangerous; if we apply custom-
ary ways of thinking to them, we obtain false results. In the ordinary word
language, we are accustomed to using the principle of interchangeability
(24-3b) implicitly. If in any of the three examples we apply this principle
to 42-6 on the basis of the true identity sentence 42-4, we obtain the
following result, 42-7. This, however, if admitted at all as a sentence, will
certainly be regarded as false.

42-7. A

B

C

Tt is not necessary
that the class Hu-
man is the same as
the class Human’.

Tt is not necessary
that the truth- value
that Scott is human
is the same as the
truth-value that
Scott is human’.

Tt is not necessary
that the individual
Walter Scott is the
same as the indi-
vidual Walter
Scott’.

These are instances of the antinomy of the name-relation in its second
form, similar to our previous example (§ 31). In spite of this result, we
may admit the formulations 42-6, provided that we are willing to prohibit
the use of the principle of interchangeability in cases of nonextensional
contexts. However, since the unrestricted use of this principle is custom-
ary and plausible, there would always be the danger of forgetting the
prohibiting rule and using the principle inadvertently. Therefore, it seems
more advisable to avoid formulations like 42-6 and, in general, formula-
tions in terms of extensions within modal or other nonextensional con-
texts.

Now let us see how the given symbolic sentences of S, are to be trans-
lated into the neutral metalanguage M'. As explained earlier, there are no
identity phrases in M'; instead, the terms ‘equivalent’ and ‘L-equivalent’
are applied in their nonsemantical use (see 34-8 and 34-9). As ‘equivalent’
is a direct translation of the symbol so is ‘L-equivalent’ of ‘s’. (This
shows again that the nonsemantical term ‘L-equivalent’ is intensional;
this holds for all nonsemantical (absolute) L-terms, see [I], § 17.) Thus
the translation of 42-1 into M' is as follows (see 34-10 and 34-13) :

190

V. ON THE LOGIC OF MODALITIES

48-8. A

B

C

‘Featherless Biped

a. ‘That Scott is a

‘The Author Of

is equivalent to Hu-

featherless biped,

Waverley is equiv-

man’.

is equivalent to

alent to Walter

that Scott is hu-
man’.

Scott’.

b. ‘Scott is a

featherless biped
if and only if
Scott is human’.

In B we add here the alternative form b because it sounds more natural

(see end of § 34).

There are two ways of translating 42-2 into M'.

The first is based on

42-2a and translates ‘N’ by ‘it is necessary that’. (In B we use again the
more natural phrase ‘if and only if’ instead of ‘is equivalent to’ ; concern-
ing the reason for the word order, see remark at the end of § 34.)

42-9a. A

B

C

‘It is not necessary

‘That Scott is a

‘It is not necessary

that Featherless Bi-

featherless biped if

that The Author

ped is equivalent to

and only if Scott is

Of Waverley is

Human’.

human, is not nec-

equivalent to Wal-

essary’.

ter Scott’.

The second alternative is based on the notation

42-2b and translates

‘ = ’ by ‘L-equivalent’ (see 34-

ii):

42-9b. A

B

C

‘Featherless Biped

‘That Scott is a

‘The Author Of

is not L-equiva-

featherless biped.

Waverley is not

lent to Human’.

is not L-equivalent

L-equivalent to

to that Scott is hu-

Walter Scott’.

man’.

This translation does not involve any difficulty analogous to that con-
nected with 42-6.

Thus the final result is as follows: It seems advisable to frame the
formulation of modal and other nonextensional sentences in the word
language, not in terms of extensions, but either (i) in terms of intensions
or (ii) in neutral terms. Which of the two formulations (i) and (ii) one
prefers is a matter of practical decision (see the discussion at the end of
§ 37). The formulation in neutral terms is simpler, but the nonsemantical

§ 43. VARIABLES IN THE WORD LANGUAGE

191

use of the terms ‘equivalent’ and ‘L-equivalent’ is not customary.
Formulations in terms of intensions, like 42-5, are, in general, more
customary, except for the reference to individual concepts in case C.
But this reference will perhaps appear less strange if we recognize the es-
sential analogy in 42-5 between C, on the one hand, and A and B, on the
other.

§ 43. Modalities and Variables in the Word Language

Translations of symbolic modal sentences with variables into M and M'
are examined. The result is analogous to that in the preceding section. It is
advisable to avoid terms of extensions and to use either terms of intensions in
M or the neutral terms in M'. The translation in terms of propositions and
properties is customary, but that in terms of individual concepts instead of
individuals may at first appear strange.

We have seen earlier (§ 10) that, as a designator has both an extension
and an intension, a variable has both value-extensions and value-inten-
sions. Therefore, a sentence with a variable can be translated into M either
in terms of its value-extensions or in terms of its value-intensions. Fur-
thermore, it can be translated into M' in neutral terms (§ 36). In analogy
to the result in the preceding section, we shall find here that it is advis-
able to avoid the formulation in terms of value-extensions and to use
either terms of value-intensions or neutral terms.

For the same reason as in the preceding section, we use here three
analogous examples, A, B, and C. They are existential sentences with the
variables ‘/’, and ‘x’ in a modal system 5 containing variables of these
types and the modal sign ‘N’.

The following sentences 43- la and b differ only in their notation. In
each of the three examples. A, B, and C, 43-ia is derived by existential
generalization from the conjunction of the sentences 42-1 and 42-2a; and
likewise 43-ib from 42-1 and 42-2b.

43-la. ABC

‘ 13 /) [(/- H) ‘( 3 /)[(/ ^ Hs) ‘( 32 :) [(2: - s)

,~N(/=H)J'. .~N(/ = Hs)]’. .~N(2: = s)]’.

43-lb. A B C

‘ ( 3 /) [ (/ - H) ‘( 3 /) [ (/ = Hs) ‘ ( 32 ;) \(x^s)

• #~(/ = Hs)]’. •~(2: = s)]’.

We shall now examine the possibilities for the translation of these sen-
tences into M. If it were a question of an extensional existential sentence
— for instance, 43-ia with the second conjunctive component omitted —
then translations in terms of value-intensions and of value-extensions

193

V. ON THE LOGIC OF MODALITIES

would be equally acceptable. This, however, is not the case for these
modal sentences. We shall first give a translation in terms of value-inten-
sions, in analogy to 42-3 and 42-5, taking notation 43-ib and translating
‘s’ by identity of intensions;

43-2. ABC

‘There is a property ‘There is a proposi- ‘There is an indi-

/ which is equivalent tion p which is vidual concept x

to but not the same equivalent to but which is equiva-

as the property Hu- not the same as the lent to but not the

man’. proposition that same as the indi-

Scott is human’. vidual concept
Walter Scott’.

In each of the three examples, this sentence can be derived by existential
generalization from the conjunction of 42-3 and 42-5.

Now we shall translate 43- la in terms of value-extensions, in analogy
to 42-4 and 42-6, translating ‘s’ by identity of extensions:

43-3. ABC

‘There is a class / ‘There is a truth- ‘There is an indi-

which is the same value p which is the vidual x which is

but not necessarily same but not nec- the same but not

the same as the class essarily the same as necessarily the

Human’. the truth-value same as the indi-

that Scott is hu- vidual Walter
man’. Scott’.

In each of the three examples, this sentence can be derived by existential
generalization from the conjunction o^f 42-4 and 42-6. We have seen in the
preceding section that formulations of modal sentences in terms of exten-
sions, like 42-6, are dangerous because they lead to the antinomy of the
name-relation unless special restrictions are imposed and that it is there-
fore advisable to avoid these formulations. The same holds for formula-
tions like 43-3.

The translation of 43-1 into neutral formulations in M', in analogy to

42- 8 and 42-9b, is as follows:

43- 4. ABC

‘There is an / such ‘There is a such ‘There is an * such
that / is equivalent that p is equiva- that x is equivalent

but not L-equiva- lent but not L- but not L-equiva-

lent to Human’. equivalent to that lent to Walter

Scott is human’. Scott’.

§ 44. QUINE ON MODALITIES

193

(Use of ‘F-equivalent’ as a nonsemantical term would provide a shorter
formulation.) In each of the three examples this sentence can be de-
rived by existential generalization from the conjunction of 42-8 and 42-9b.
The formulations 43-4 are free of the dangers involved in 43-3.

Now let us compare the three examples, A, B, and C. Our proposal not
to translate variables in modal sentences in terms of extensions seems
quite natural in cases B and A. As remarked earlier (§ 40), it seems that
all logicians interpret modal sentences in terms of propositions rather than
of truth-values, and most of them use terms of properties rather than of
classes. Only in case C does our interpretation deviate from the custom-
ary one. The reference to individual concepts may first appear some-
what strange; and the alternative translation in neutral terms (e.g.,
43-4C), which avoids the reference to individual concepts, uses the un-
familiar terms ‘equivalent’ and ‘L-equivalent’. However, I believe that,
once we are aware of the perfect analogy between the three cases, we
recognize the inadequacy of the formulations in terms of individuals; and
the impression of strangeness which the formulation in terms of individual
concepts and, to a lesser degree, the neutral formulation may first give will
perhaps disappear. Modal sentences with variables are of a quite peculiar
logical nature, and it should not be surprising that an adequate and cor-
rect rendering for them in the word language is not always possible in
entirely customary and natural terms.

§ 44 . Quine on Modalities

Quine’s article [Notes] explained his view that, under customary conditions,
modalities and quantification cannot be combined. A new statement by Quine
is quoted here, in which he says that my language succeeds in combining
modalities with quantification but only at the price of repudiating all exten-
sions, for instance, classes and individuals. I try to show that my modal lan-
guage does not exclude anything that is admitted by a corresponding exten-
sional language.

Quine® illustrates the difficulty which we have called the antinomy
of the name- relation by the following example among others (as men-
tioned above, § 31). We find as an arithmetical and hence logical truth;

(i) ‘9 is necessarily greater than 7’.

The following is a true statement of astronomy :

(ii) ‘The number of planets = 9’.

5 Quine [Notes] (i8) p. 121, (15) p, 119, (23) p. 121.

194

V. ON THE LOGIC OF MODALITIES

If, in (i), ‘9’ is replaced by ‘the number of planets’ in virtue of the true
identity statement (ii), we obtain the false statement:

(iii) ‘The number of planets is necessarily greater than 7’.

Quine’s method for solving the antinomy has been explained earlier
(§32, Method II). According to our method, the following sentence takes
the place of (ii) in M':

(iv) ‘The number of planets is equivalent to 9’.

The sentences (i) and (iii) occur also in M'. But now it is not possible to
infer the false sentence (iii) from the true sentence (i) together with (iv) .
According to the first principle of interchangeability (i2-i), the expres-
sions ‘the number of planets’ and ‘9’ are interchangeable on the basis of
(iv) in extensional contexts only, hence not in (i). Thus the difficulty
disappears, and the designators occurring in nonextensional contexts still
function, according to our conception, as normal designators.

An even more serious problem is raised by Quine’s objection to modal
sentences with variables. He discusses the following expression :

(v) ‘There is something which is necessarily greater than 7’.

He says® that this expression “is meaningless. For, would 9, that is, the
number of planets, be one of the numbers necessarily greater than 7?
But such an affirmation would be at once true in the form , . . (our
(i)] and false in the form . . . [our (iii)].” Quine does not regard (i) and
(iii) as meaningless. As explained earlier (§32, Method II), he regards
occurrences of designators in nonextensional contexts, e.g., ‘9’ in (i) and
‘the number of planets’ in (iii), as “not purely designative” ; in other
words, these occurrences do not function as names, and hence the principle
of interchangeability is not applicable. For the same reason, according to
Quine’s view, the rule of existential generalization is not applicable to
these occurrences. Therefore, there is no valid inference from (i) to (v),
and, moreover, (v) has no meaning and hence cannot be admitted as a
sentence. Thus Quine arrives at the following conclusions, which are
stated at the end of his paper: “A substantive word or phrase which desig-
nates an object may occur purely designatively in some contexts and not
purely designatively in others. This second type of context, though not
less “correct” than the first, is not subject to the law of substitutivity of
identitynor to the laws of application and existential generalization. More-
over, no pronoun (or variable of quantification) within a context of this

" Ibid., p. 124.

§44. QUINE ON MODALITIES 19 s

second type can refer back to an antecedent (or quantifier) prior to that
context. This circumstance imposes serious restrictions, commonly un-
heeded, upon the significant use of modal operators, as well as challeng-
ing that philosophy of mathematics which assumes as basic a theory of
attributes [i.e., properties] in a sense distinct from classes.”’

To Quine’s contexts of the second kind belong all those which we call
nonextensional. He discusses, in particular, contexts within quotes and
modal contexts. With respect to contexts within quotes his conclusions
are no doubt correct. I cannot agree, however, with Quine’s conclusion
concerning modal contexts. We have combined modalities and variables
both in symbolic object languages (§ 40) and in word formulations in our
metalanguages (§43).

Church likewise does not accept Quine’s result. He says in the review
of Quine’s paper that he “would question strongly the conclusion which
the author draws that no variable within an intensional context . . . can
refer back to a quantifier prior to that context .... The conclusion
should rather be that in order to do this a variable must have an inten-
sional range — a range, for instance, composed of attributes [properties]
rather than classes.”* Up to this point I am in agreement with Church.
His solution is as follows: He distinguishes, like the system PM (see § 27),
between class variables, e.g., ‘a’, and property variables, e.g., V. He takes
as example a sentence which is essentially the same as a conjunction of
42-iA and 42-2aA. In distinction to Quine, he regards it as admissible to
infer from this sentence by existential generalization an existential sen-
tence; the latter, however, must not have the form ‘( 3 a) (. . a . .)’ but
rather the form ‘( 3 <^)(. . 0 . It seems to me that this procedure is
correct and, indeed, solves completely the difl&culty pointed out by Quine.
I believe, however, that there is a simpler way to achieve this. It is simi-
lar to that of Church but avoids the use of two kinds of variables for the
same type. This use is, as explained earlier (§ 27), an unnecessary duplica-
tion. It is sufficient to use variables of one kind which are neutral in the
sense that they have classes as value-extensions and properties as value-
intensions; this is done in 43-iaA. The use of different variables for exten-
sions and intensions within all types would lead in the case of Quine’s
example (v) to the introduction of variables for number concepts different
from the variables for numbers. This, however, would be both unneces-
sary and unusual.

The problem of whether or not it is possible to combine modalities and

''Ibid., p. 127.

“ [Review Q.], p. 46.

196 V. ON THE LOGIC OF MODALITIES

variables in such a way that the customary inferences of the logic of quan-
tification — in particular, specification and existential generalization — re-
main valid is, of course, of greatest importance. Any system of modal
logic without quantification is of interest only as a basis for a wider sys-
tem including quantification. If such a wider system were found to be im-
possible, logicians would probably abandon modal logic entirely. There-
fore, it is essential to clarify the situation created by Quine’s analysis and
objections. For this reason I have asked Quine, who has read an earlier
version of the manuscript of this book, for a statement of his present view
on the problem mentioned and, in particular, his reaction to my method
for combining modalities and variables as explained in the preceding sec-
tion. With his kind permission, I am quoting here his statement in full :*

Every language system, insofar at least as it uses quantifiers, assumes one or
another realm of entities which it talks about. The determination of this realm
is not contingent upon varying metalinguistic usage of the term ‘designation’
or ‘denotation’, since the entities are simply the values of the variables of
quantification. This is evident from the meaning of the quantifiers '(»)’, '(/)’,
'{PY, ‘(a/)’, '(af')’ themselves: ‘Every (or. Some) entity * (or/ or p) is

such that’. The question wkat there is from the point of view of a given language
— the question of the ontology of the language — is the question of the range
of values of its variables.

Usually the question will turn out to be in part an a priori question regard-
ing the nature and intended interpretation of the language itself, and in part
an empirical question about the world. The general question whether for ex-
ample individuals, or classes, or properties, etc., are admitted among the values
of the variables of a given language, will be an a priori question regarding the
nature and intended interpretation of the language itself. On the other hand,
supposing individuals admitted among the values, the further question whether
the values comprise any unicorns will be empirical. It is the former type of in-
quiry — ontology in a philosophical rather than empirical sense — that interests
me here. Let us turn our attention to the ontology, in this sense, of your
object language.

An apparent complication confronts us in the so-called duality of M' as be-
tween intensional and extensional values of variables; for it would appear then
that we must inquire into two alternative ontologies of the object language.
This, however, I consider to be illusory; since the duality in question is a
peculiarity only of a special metalinguistic idiom and not of the object language
itself, there is nothing to prevent our examining the object language from the
old point of view and asking what the values of its variables are in the old-
fashioned non-dual sense of the term.

It is now readily seen that those values are merely intensions, rather than
extensions or both. For, we have:

(x)(x = x),

i.e., every entity is L-equivalent to itself. This is the same as saying
that entities between which L-equivalence fails are distinct entities — a

* The first two-thirds of Quine’s statement as here quoted is dated October 23, 1945; the
remainder January i, 1946.

§ 44 . QUINE ON MODALITIES

197

clear indication that the values (in the ordinary non-dual sense of the term)
of the variables are properties rather than classes, propositions rather than
truth-values, individual concepts rather than individuals. (I neglect the further
possibility of distinctness among L-equivalent entities themselves, which would
compel the entities to be somehow “ultra-intensional”; for it is evident that
you have no cause in the present connection to go so far.)

I agree that such adherence to an intensional ontology, with extrusion of
extensional entities altogether from the range of values of the variables, is in-
deed an effective way of reconciling quantification and modality. The cases of
conflict between quantification and modality depend on extensions as values of
variables. In your object language we may unhesitatingly quantify modalities
because extensions have been dropped from among the values of the variables;
even the individuals of the concrete world have disappeared, leaving only their
concepts behind them.

I find this intensional language interesting, for it illustrates what it would be
like to be able to give the modalities free rein. But this repudiation of the con-
crete and extensional is a more radical move, in general, than a mere compari-
son of 43-3 with 43-2 might suggest. The strangeness of the intensional language
becomes more evident when we try to reformulate statements such as these;

(1) The number of planets is a power of three,

(2) The wives of two of the directors are deaf.

In the familiar logic, (i) and (2) would be analyzed in part as follows;

(3) (3«) (m is a natural number . the number of planets = 3") ,

(4) (3*)(3y)(3s)(3w)[* is a director . y is a director , ~ (* = y) . z is
wife of X . ui is wife of y . z is deaf . w is deaf).

But the formulation (3) depends on there being numbers (extensions, presum-
ably classes of classes) as values of the bound variable; and the formulation (4)
depends on there being persons (extensions, individuals) as values of the four
bound variables. Failing such values, (3) and (4) would have to be reformu-
lated in terms of number concepts and individual concepts. The logical predi-
cate ‘ = ’ of identity in (3) and (4) would thereupon have to give way to a logical
predicate of extensional equivalence of concepts. The logical predicate ‘is a
natural number’ in (3) would have to give way to a logical predicate having the
sense ‘is a natural-number-concept’. The empirical predicates ‘is a director’,
‘is wife of’, and ‘is deaf’, in (4), would have to give way to some new predicates
whose senses are more readily imagined than put into words. These examples do
not prove your language-structure inadequate, but they give some hint of the
unusual character which a development of it adequate to general purposes
would have to assume.

The first important point to be noticed in Quine’s statement is that he
agrees that the form of modal language explained in the present chapter
“is indeed an effective way of reconciling quantification and modality”.
Some readers of Quine’s article believed that it proved the impossibility
of a logical system combining modalities with variables. Quine’s statement
now shows that this is not the case.

However, there are still some serious problems involved. Quine, while
admitting the possibility of modal systems with quantification, believes

198 V. ON THE LOGIC OF MODALITIES

that these systems have certain peculiar features which he regards as
disadvantages. Let us now examine these problems.

I have previously explained (at the beginning of § lo) that I agree with
Quine’s view that an author who uses variables of some kind thereby
indicates that he recognizes those entities which are values of the vari-
ables. (I have simultaneously expressed some doubts concerning the ad-
visability of applying the term ‘ontology’ to this recognition ; but for our
present discussion we may leave aside this question.) It is the counterpart
of this thesis that is of importance for our problem; it says that, if some-
one uses a language which does not contain any variables with certain
entities as values, he thereby indicates that he does not recognize these
entities or at least that he does not intend to speak about them as long as
he restricts himself to the use of this language. In a certain sense, I can
agree also with this thesis. As an example, let us compare the following
two languages S? and Sp. Let Sp be the ordinary language of physics
(§ 19) . It contains variables which have real numbers, both rational and
irrational, as values. Suppose somebody proposes another language Sp
for physics which contains variables for rational numbers, but no vari-
ables to whose values irrational numbers belong. Here I would be willing
to say, like Quine, that the user of this language Sp excludes or “repudi-
ates” the irrational numbers and that these numbers “have disappeared”
from the universe of discourse. Now Quine says that the variables in the
modal language have as values only intensions, not extensions, and that
therefore, as far as this language is concerned, all extensions, for example,
classes and “the individuals of the concrete world”, “have disappeared”.
With this I cannot agree. At the first glance, the situation here may seem
to be similar to that in the example of the irrational numbers; but actually
it is fundamentally different.

In order to clarify the situation, we shall contrast in the following dis-
cussion our two language systems, the extensional language S, and the
modal language Sa. We shall further consider the following two extended
languages. The language S, is extensional like S, but contains additional
kinds of variables, say ‘g’, etc., for which predicators of level one (and
degree one) are substitutable, ‘m’, ‘w’, etc., for predicators of level two,
and ‘p’, ‘g’, etc., for sentences. The language Sj is constructed from S,' by
the addition of ‘N’; hence it is a modal language like Sj. According to
Quine’s view, the values of ‘/’ in S, are not classes but properties, because
‘(f)(/ ^ /)’ holds. In the extensional system S,, on the other hand, we
have only ‘(f) (/ = /)’. Therefore, Quine will presumably regard classes
as the values of ‘/’ in this system, as he does for the variables of his ex-

§ 44. QUINE ON MODALITIES

199

tensional system ML (see above, § 25). Similarly, Quine says that the
values of individual variables (e.g., ‘x’) in modal systems like Sa and S,
are individual concepts; on the other hand, he presumably regards indi-
viduals (concrete things or positions) as the values of individual variables
in extensional systems like S, and S*. Now the decisive point is the follow-
ing: As explained previously (§ 35), there is no objection against regard-
ing designators in a modal language as names of intensions and regarding
variables as having intensions as values, provided we are not misled by
this formulation into the erroneous conception that the extensions have
disappeared from the universe of discourse of the language. As explained
earlier (§ 27), it is not possible for a predicator in an interpreted language
to possess only an extension and not an intension or, in custpmary terms,
to refer only to a class and not to a property. Similarly, it is impossible for
a variable to be merely a class variable and not also a property variable.
On the other hand, it is, of course, possible for a variable to have as values
only properties and no relations, or only rational numbers and no irra-
tional numbers. This shows the difference between the two cases. For
example, the so-called class variables in the system PM' (e.g., ‘a’) are, as
we have seen (§27), also property variables, that is to say, they have
properties as value intensions. The same holds now for variables like ‘f
in s’. Languages of Quine’s form ML' or of Russell’s form PM' or of our
form S, speak also about properties. The restriction of these extensional
languages in comparison with modal languages like S, consists merely in
the fact that whatever is said in any of these languages about a property
is either true for all equivalent properties or false for all equivalent
properties; in technical terms, all properties of properties expressible in
these languages (by a matrix with a free variable of the kind mentioned)
are extensional. This makes it possible to paraphrase all sentences of
these languages in terms of classes. An analogous result holds for indi-
vidual variables. These variables in an extensional language like S, and
s’ refer not only to individuals but also, and even primarily, to individual
concepts. The restriction is again merely this: Whatever is said in these
languages about individual concepts is either true for all equivalent indi-
vidual concepts or false for all of them; in technical terms, it is extensional.
Therefore, whatever is said in these languages about individual concepts
can be paraphrased in terms of individuals.

Although the sentences of an extensional language (S, or SJ can thus
be interpreted as speaking about individuals and classes, they can be
translated into the corresponding modal language (Sj or S’, respectively).
This translation fulfils not only the requirement of L-equivalence but

300

V. ON' THE LOGIC OF MODALITIES

also the requirement of intensional isomorphism, the strictest requirement
that any translation can fulfil (§ 14). Any given sentence in S, is trans-
lated into S, by that sentence itself, that is, by the same sequence of signs
now taken as signs in S’. Any two corresponding designators, that is, any
designator in S, and the same expression in S,, are L-equivalent to one
another. This follows from the following two results:

(i) The rules of designation for the descriptive signs are the same in
both systems S, and (for example, the rules 1-2 for primitive predica-
tors).

(ii) Any sentence in S, has the same range in both systems S and S,
(see 41-4 concerning S. and Sj). Since the range is the same, the truth-
conditions are the same ; therefore, the sentence means exactly the same
in Sj as in S,*.

Thus the decisive difference between the situation here and that in the
earlier example concerning the irrational numbers becomes clear. In the
transition from Sp to Sp the irrational numbers actually disappear, be-
cause a sentence in Sp of the form ‘there is an irrational number such
that . . .’ is not translatable into Sp. On the other hand, in the transition
from an extensional to a modal language the individuals and classes do by
no means disappear. A sentence in S, (or S,’) which says that there is an
individual of a certain kind is translatable into Sa (or S^) ; and a sentence
in S, which says that there is a class of a certain kind is translatable into

s;.

In order to illustrate this result by an example, let us take Quine’s
sentence (2) . Since this sentence requires only individual variables, it can
be translated into Si. Let us assume that Si contains the following predica-
tors, either as primitive signs or as defined in a suitable way : ‘ W’ for the
relation Wife, ‘D’ for the property Director, and ‘F’ for the property
Deaf. Then (2) is translated into Si by the following sentence:

(5) ‘(3a;)(ay)(3z)(3M>)[D2; • Dy • '^(x = y) • Wzx • Wwy • Fz • Fw]’.

Now this same sentence is also the translation of (2) into Sa. It would be
an error to think that it was necessary for the translation into Sj either
to use new predicators or to assign a new meaning to the old predicators,
as though, for example, ‘Da:’ in Si said that the individual x has the prop-
erty Director while ‘Da:’ in Sa said that the individual concept x has a
strange new property somehow analogous but not quite the same as the
property Director. The matrix ‘Da:’ expresses in both languages the
property Director; it may be defined in both languages in exactly the same

§ 44. QUINE ON MODALITIES

aoi

way. Suppose a speaker X, uses the language Si and X, uses S,. Then the
question of whether a given full sentence, say ‘Db’, is true, may be de-
cided by both speakers in the same way. Both confirm or disconfirm this
sentence on the basis of observations of the person b, using the same
empirical criteria for the property Director. Nothing in the semantical
analysis of this sentence or in the procedure of empirical confirmation or
in the expectation of possible future experiences implied by the sentence
needs to be different for the two speakers. The same holds for the existen-
tial sentence (5) and for any other sentence occurring in both languages.
Therefore, I cannot agree with the view that, while the speaker X,
recognizes the individuals of the concrete world, they have disappeared
for Xi, leaving only their concepts behind them.

The situation with respect to Quine’s other example (i) is analogous,
except that cardinal numbers are involved and therefore a variable of
second level, say is used. We have seen earlier (§ 27) that, for the in-
troduction of particular cardinal numbers and of the general concept of
cardinal number, it is not necessary to use special class expressions and
class variables, as Frege and Russell did; we may, instead, regard cardinal
numbers as properties of second level or, rather, introduce cardinal num-
ber expressions as predicators of second level, whose intensions are proper-
ties of second level and whose extensions are classes of second level. Equal-
ity of cardinal numbers is then expressed with the help of “ = Thus we
translated the sentence

(6) ‘the number of planets = 9’
into the following sentence of S, :

(7) ‘Nc‘P ^ 9’.

Similarly, Quine’s sentence (i) can be translated into S,’ as follows, if we
assume that exponentiation has been defined by a suitable procedure
(analogous to that of Cantor or Russell, [P.M.], Vol. II, *116):

(8) ‘( 3 »)[NC(w) .Nc‘P = 3"]’.

(If we wish to say that n is finite, we may use the concept of inductive
cardinal number with a definition analogous to Russell’s). Here, again, the
given sentence (i) can likewise be translated into the modal language Sj,
namely, by the same sentence (8), hence without the use of any strange
new concepts. The translation is by no means dependent upon the occur-
rence of class variables as distinct from property variables. ‘NC(n)’
means in S,, just as in S(, that « is a cardinal number; thus in S,’, just
as in S(, sentences like ‘NC(2)’ and ‘NC(Nc‘P)’ are L-true. That the

202

V. ON THE LOGIC OF MODALITIES

sentence ( 8 ) has in Sj the same factual content as in S’ is seen by con-
siderations similar to those concerning the previous example ( 5 ). The same
astronomical observations confirm the sentence in the one as in the other
language; it gives rise to the same expectations of future observations in
both languages. Thus there cannot be any difference in meaning.

The preceding discussion shows that a modal language is not inadequate
in comparison with the corresponding extensional language, that is to
say that we can express in the former whatever is expressed in the latter.
(So much Quine seems to admit.) We have seen, moreover, that the ex-
pressions used in a modal language for translations from the extensional
language do not have any unusual character with respect to either their
form or their meaning. Every designator and every sentence in the exten-
sional language has exactly the same meaning in the modal language-
more exactly speaking, it has both the same intension and the same exten-
sion. The world of concrete things and the conceptual world of numbers
are dealt with in the modal language just as well as in the extensional one.
In order to see correctly the functions of these languages, and generally
of any languages, it is essential to abandon the old jjrejudice that a predi-
cator must stand either for a class or for a property but cannot stand for
both and that an individual expression must stand either for an indi-
vidual or for an individual concept but cannot stand for both. To under-
stand how language works, we must realize that every designator has both
an intension and an extension.

§ 46. Conclusions

The main conclusions of the discussions in this book are briefly summarized.
The difference between the two operations — understanding the meaning of a
given expression and investigating whether and how it applies to the actual
state of the world — suggests a distinction between two different semantical
factors, which our method tries to explicate by the concepts of the intension and
the extension of an expression.

The chief purpose of this book is to develop a method for the analysis
of meaning in language, hence a semantical method. We may distinguish
two operations with respect to a given linguistic expression, in particu-
lar, a (declarative) sentence and its parts. The first operation is the
analysis of the expression with the aim of understanding it, of grasping its
meaning. This operation is a logical or semantical one; in its technical
form it is based on the semantical rules concerning the given expression.
The second operation consists in investigations concerning the factual
situation referred to by the given expression. Its aim is the establishment

§45. CONCLUSIONS

203

of factual truth. This operation is not of a purely logical, but of an em-
pirical, nature. We can distinguish two sides or factors in the given ex-
pression with regard to these two operations. The first factor is that side
of the expression which we can establish by the first operation alone, that
is, by understanding without using factual knowledge. This is what is
usually called the meaning of the expression. In our method it is expli-
cated by the technical concept of intension. The second factor is estab-
lished by both operations together. Knowing the meaning, we discover
by an investigation of facts to which locations, if any, the expression ap-
plies in the actual state of the world. This factor is explicated in our
method by the technical concept of extension. Thus, for every expression
which we can understand, there is the question of meaning and the ques-
tion of actual application; therefore, the expression has primarily an in-
tension and secondarily an extension.

The method of intension and extension stands in contrast to the
customary method of the name-relation. The basic weakness of the latter
method is its failure to realize the fundamental distinction between mean-
ing and application. This leads to the conception that an exj)ression must
be the name of exactly one of the two semantical factors involved. For
example, properties and classes are regarded as entities of equal standing;
this leads to the view that a language ought to contain both names of
properties and names of classes. This conception is the ultimate source
of the various difficulties which we found involved in the method of the
name-relation. They center around the well-known difficulty which we
have called the antinomy of the name-relation. We have seen how the
various methods of keeping the name-relation but avoiding the antinomy
lead either to great complications in the language structure or to serious
restrictions in the use of the language or in the application of the semanti-
cal method.

The formulations in terms of ‘extension’ and ‘intension’, ‘class’ and
‘property’, .etc., seem to refer to two kinds of entities in each type. We
have seen, however, that, in fact, no such duplication of entities is presup-
posed by our method and that those formulations involve only a con-
venient duplication of modes of speech. As it was shown to be unnecessary
to use different expressions for classes and properties in a symbolic object
language, it likewise turned out to be unnecessary to use those pairs of
terms in the word language as a metalanguage. A new metalanguage was
constructed, in which instead of the pair of phrases ‘the class Human’ and
‘the property Human’ only the neutral term ‘Human’ is used. It was
shown that the ordinary formulations can be translated into this neutral

204

V. ON THE LOGIC OF MODALITIES

metalanguage and that the latter language preserves all previous distinc-
tions, though in different formulations.

Our semantical method also helps in the clarification of the problems of
the modalities. It suggests a certain interpretation of the logical modali-
ties which supplies a suitable basis for a system of modal logic. In par-
ticular, the distinction between intensions and extensions enables us to
overcome the difliculties involved in combining modalities with quantified
variables.

The different conceptions of other authors discussed in this book, for
instance, those of Frege, Russell, Church, and Quine, concerning semanti-
cal problems, that is, problems of meaning, extension, naming, denotation,
and the like, have sometimes been regarded as different theories so that
one of them at most could be right while all others must be false. I regard
these conceptions and my own rather as different methods, methods of
semantical analysis characterized chiefly by the concepts used. Of course,
once a method has been chosen, the question of whether or not certain re-
sults are valid on its basis is a theoretical one. But there is hardly any
question of this kind on which I disagree with one of the other authors.
Our differences are mainly practical differences concerning the choice of
a method for semantical analysis. Methods, unlike logical statements, are
never final. For any method of semantical analysis which someone pro-
poses, somebody else will find improvements, that is, changes which will
seem preferable to him and many others. This will certainly hold for the
method which I have proposed here, no less than for the others.

Let me conclude our discussions by borrowing the words with which
Russell concludes his paper.'® It seems to me that his remarks, although
written more than forty years ago, still apply to the present situation
(except, perhaps, that instead of ‘the true theory’ I might prefer to say
‘the best method’) :

“Of the many other consequences of the view I have been advocating,
I will say nothing. I will only beg the reader not to make up his mind
against the view — as he might be tempted to do, on account of its ap-
parently excessive complication — until he has attempted to construct
a theory of his own on the subject of denotation. This attempt, I believe,
will convince him that, whatever the true theory may be, it cannot have
such a simplicity as one might have expected beforehand.”

'« [Denoting], p. 493.

BIBLIOGRAPHY

[The abbreviated titles in brackets are used in citations throughout this book.]

Bennett, Albert A., and Bavlis, Charles A. [Logic]. Formal Logic: A Modern
Introduction. New York, 1939.

Carnap, Rudolf. [Syntax]. Logical Syntax of Language. Orig., Vienna, 1934; English
trans. London and New York, 1937.

. [I]. Introduction to Semantics. “Studies in Semantics”, Vol. I. Cambridge,

Mass., 1942.

-. [II]. Formalization of Logic. “Studies in Semantics”, Vol. II. Cambridge,
Mass., 1943.

. [Inductive]. “On Inductive Logic”, Philosophy of Science, XII (1945), 72-97.

-. [Remarks]. "Remarks on Induction and Truth”, Philosophy and Phenom-
enological Research, VT (1946), 590-602.

-. [Modalities]. “Modalities and Quantification”, Journal of Symbolic Logic, XI
(1946). 33-64-

Church, Alonzo. [Dictionary]. Articles in D. D. Runes (ed.), The Dictionary of
Philosophy. New York, 1942.

. [Review C.]. “Carnap’s Introduction to Semantics” (a review of Carnap, [I]),

Philosophical Review, LII (1943), 298-304.

. [Review Q.]. A review of Quine, [Notes], Journal of Symbolic Logic, VTII

(1943). 4S-47-

Frege, Gottlob. [Sinn]. “Ueber Sinn und Bedcutung”, Zeilschrift fur Philosophic
und philosophische Kritik, C (new ser., 1892), 25-50.

. [Grundlagen]. Die Grundlagen dcr Arithmetik: Bine logisch-mathematische

Untersuchung iiber den Begrif der ZaM. Breslau, 1884. Reprinted, 1934.

-. [Grundgesetze]. Grundgrsetze dcr Arithmetik, Vols. I, II. Jena, 1893; 1903.

Hilbert, David, and Bernavs, Paul. [Grundlagen I]. Grundlagen der Mathematik,
Vol. I. Berlin, 1934.

Lewis, C. I. A Survey of Symbolic Logic. Berkele}', 1918.

. [Meaning]. “The Modes of Meaning”, Philosophy and Phenomenological Re-
search, IV (1943-44), 236-50.

Lewis, C. I., and Langford, C. H. Symbolic Logic. New York, 1932.

Morris, Charles. [Signs]. Signs, Language, and Behavior. New York, 1946.

Quine, W. V. [Designation]. “Designation and Existence”, Journal of Philosophy,
XXXVI (1939), 70I-9-

— . [M.L.]. Mathematical Logic. New York, 1940.

. [Notes]. “Notes on Existence and Necessity”, Journal of Philosophy, XL

(1943), 113-27-

Russell, Bertrand. [Denoting]. “On Denoting”, Mind, XIV (new ser., 1905), 479-
93-

— . [P.M.]. See Whitehead.

. [Inquiry]. An Inquiry into Meaning and Truth. New York, 1940.

Tarski, Alfred. [WahrheitsbegrifI]. “Der Wahrheitsbegriff in den formalisierten
Sprachen”, Studia philosophica, I (1936), 261-405. Originally published in 1933.

. [Truth]. “The Semantic Conception of Truth and the Foundations of Seman-
tics”, Philosophy and Phenomenological Research, IV (1944), 341-76.

Whitehead, A. N., and Russell, B. [P.M.]. Principia Mathematica. 3 vols. Cam-
bridge, England, 1910-13. 2d ed., 1925-27.

Wittgenstein, Ludwig. [Tractatus]. Tractatus logico-philosophicus. London, 1922.

205

INDEX

cateX” i“f«rtant terms, names, and reference, are indi

A, a, 3 , 4 Cognitive meaning, 6

.. Computable numbers, 7o n .

“ f -i/

‘Absolute' use of terms, 24 n.; see also Non-
semantical use
Abstraction expressions, 3
Actual and nonactual things, 65
Adequacy: for F-concepts, 12; for L-con-
cepts, Ilf.; forL-truth, lof.

Ambiguity of the name-relation, 100 £f.
Analysis, paradox of, 63
Analytic sentences, 8; see also L-truth
Antinomies, nature of, 135
Antinomy of the name-relation, 133 ff., i8o‘
solutions for, 136 £f. ’

Arithmetical concepts, 84 1 ., 90, i ij ff.

Atomic sentences, <;
au, 187

.Autonyraous use, 4
Axiom of infinity, 86 n.

B, 4

Baylis, C. a., 21 n,, 28, 64 n., 205
Belief-sentences, 53 ff,, 61 f,, 123!., 141
Bennett, A. A., 21 n„ 28, 205
BeRNAYS, P., 33, 38, 205
Biconditional, 3
Black, M., 63
Blanks, 4

Cantor, G., 201
Capitalization, 17 n., 153
Cardinal number, 1 16, '201; see also Number
CHTTRCH, A. : Bibliography by, 1 73 n . ; on de-
noting, 97 n.; ‘description,’ 126 n.; on
designatum, son., 166 n.; explicanda of,
127; on extensionality, 470., son.; on
Frege, 118, 125, 126, 132, 137 f.; infinity
of names, 130 138; on lambda-de-

finability, 79 n.; on lambda-operator,
39 n.; method of, of name-relation, 137 f.,
143, 204; on modal sentence, 187; publica-
tions of, 3, 205; on Quine, 100, 137, 195;
results of, are valid, 128; on Russell, 140;
on synonymity, 99; on two abstraction
operators, no; on two kinds of variables

Llass names, 107, in f.

Class variables, 113, 160, 201
Classes, 16 ff., contextual definition of, 114,
146, 147. 159; extensions, 16 ff.; as in-
tensions, 92; Russell’s definition of, 107
Classes of sentences, 178 f.

Closed sentences, 3

Concept, 21
Conditional, 3
Conjunction, 3
Connectives, 3
Connotation 65, it8 n., 126
Contingent propositions, 175
Contingent truth, 12, 132
Conversion, 3

Coordinate languages, 75, 79

CT, 132

Degree, 5
Denotation, 126
Denoting, 97 n., n8 n.

Describing an extension, 71
De^ription: {:) in^vidual, 3, 32 ff.; in S,,
184 f.; (2) other types, 38 f.

Descriptive expressions, 3, 86
Descriptum, 33

Designation, 97, 99, 162 f., 171
Designators, 6 f.

‘Designatum,’ i66n.

Disjunction, 3

Ducasse, C. J.: articles by, 640.; on epis-
temic attitudes, 62; on facts, 28
Duplication: of names, 106 ff.; on higher
levels, ii4f.; of variables, 113

Eaton, R. M., 126 n.

Empty predicators, 21

Entitks, 20, 23 , 43 1 ., 91, 146, 167!, 196;

see also Multiplicity of entities
Enumeration, 83, nr
Equality of numbers, 1 1 7
Equinumerous properties, 116
Equivalence: (i) semantical, of expressions:
of designators, 14; of individual expre.s-
sions, is; of predicators, 14 f.; of sen-
tences, 6; (2) nonsemantical, of entities:
of extensions, 25; of intensions, 24; of
neutral ^titles, 154; of properties, 24; of
propositions, 24

Equivalence, logical; see L-equivalence
Equivalence class, 16
Exemplification, 21
Existence, 43

Explicandum, explication, explicatura, 8
Extension: concerns application, 157, 203; of
designators, 23; of individual expressions,

^ 40 J of predicators, 18 f.; of sentences, 26
Extensional expressions, 47 f.

20 &

INDEX

Extensional language: avoids antinomy,
141 1.; definition of, 48; refers to proper-
ties, 199; for semantics, 168 S.

Extensional metalanguage, 169
Extensionality, thesis of, 141

F, 4

/, 38. 44 f-
F-concepts, 12

F-equivalence: of designators, 14; of indi-
vidual expressions, 40; of predicators, 15;
of sentences, la
F-falsity, la
F'implication, la
F-truth, la
Fact, 12, a8
Factual sentences, la
Falsity, 6

FREGE, G.: on antinomy of name-relation,
1^3, 136; on belief-sentences, 123 f., 132; in
bibliography, aos; on descriptions, 8, 34,
as B-i 38, 140; disadvantages of method of,
129 ff., I37i 1431 on double nominatum,
13a; explicanda, lafif.; Infinity of names
and entities, 130 f.; on nominatum,
iigf., J2s; on number, 8, 116 f., 201; on
oblique context, 123; principles of inter-
changeability, 99, lai f., 143; on ‘propo-
sition,’ ii8n., 124; results are valid, 128;
semantical method of, 119, 204; on sense,
1 19, 125; terminology of, 118 n.

Functors, 7

G, 101

German letters, 4, 72
Giving an extension, 71
Goedel, K,, 36, 79 n.

Gooduan, N., 36 n.

H, 4

Herbrand, J,, 79 n.

Hilbert, D., 33, 34, 38, 205
Hold, 9, 170
Hypostatization, aa, 44

Identity, 14

Identity condition: for classes, 18; for ex-
tensions, 23; for intensions, 23; for proper-
ties, 18; for propositions, 27, 124
Identity expression, sentence, sign, 99
Implication, logical; see L-implication
Implication, material; see Conditional
Impossibility, 175
Impossible things, 65
Individual concepts, 41, 180 f.

Individual constants, 4, 32
Individual description; see Description (1)
Individual expressions, 3a
Individual variables, 3, 32
Individuals, 32; as numbers, 86; as positions,
74

Inductive logic, 9 n .

Intension: of designators, 23; of individual
expressions, 41; of predicators, 18 f.; as
primary, tia, 157, 203; of sentences, 27

Intensional expressions, 48
Intensional isomorphism, 56, 59
Intensional structure, 36
Interchangeability, 47; principles of, 51 f.
Interpretation, s, 9
Iota-operator, 3, 9, 3a

K, 112

Kant, I., 8, 12
Kleene, S. C., 79 n.

L-concepts, 7ff.

L-designation, 163

L-determinacy: (i) semantical: of designa-
tors, 70, 72, 88; of functors, 85; of indi-
vidual expressions, 78, 80; of predicators,
83; of sentences, 11; (2) nonsemantical :
of intensions, 89
L-empty predicators, ai, 84
L-equivalence: (i) semantical, of expres-
sions; of designators, 14; of individual
expressions, 40; of matrices, 58; of predi-
cators, 15; of sentences, ii; in two sys-
tems, S7; (2) nonsemantical, of entities;
of extensions, 25; of intensions, 24 f,; of
neutral entities, 154; of properties, 25; of
propositions, 24
L-cquivalence class, 16, 152
L-falsity, ii, 31
L-implication, ii

L-indeterminacy: of designators, 72; of
sentences, 12
L-interchangeability, 47
L-range; see Range
L-state; see State
L-truth, 8, 10
L-universality, 44, 84
Lambda-operator, 3, 9, 39
Langford, C. H., 8n., 205; on analysis,
631.

Language of physics, 79
Leibniz, G. W., 8, 9, 10
Leonard, H. S., 36 n.

LEWIS, C. I.: in bibliography, 205; on com-
prehension, connotation, denotation, 64 f.;
definition of necessity, 17s; on denotation
of true propositions, 94 f.; on equivalence
in analytic meaning, 60 f., 62; on modal
logic, 173; on nonactual things, 65!.; on
propositions, 179; on strict equivalence,
177; on strict implication, 47 n., 177; on
sjmonymity, 60 f.

Logical expressions, 86
Logical truth; see L-truth

M, 4. MS
M', 153, 167
M., 169

Martin, R. M., 36 n.

Material implication; see Conditional
Mathematical concepts; see Arithmetical
concepts
Matrix, s

Meaning, 6 f,, 10, 60, 63, 64, 1 18 n., 202 f.

INDEX

309

Meinong, a. von, on impossible things, 65
Mental entities, 20
Metalanguage, 4, 44
Metametalanguage, 71, 153
Metaphysics, 43

Methods: of extension and intension, 2, 33,
107, 143 f., 167, 303 f.; of the name-rela-
tion, p8, 144, aoj; comparison of the two,
106 f., 109 f., 117, 131, 143
MFL, 183 n.

Miii, John Stuart, 126
Mixed modalities, 175

ML, 103
ML', 103, 199

MM, I S3

Modalities: analogous to quantifiers, 186;
correspondence to L-concepts, 174 f.;
definitions of, 175; intensional, 50; inter-
pretation of, 176, 178; Lewis on, 173;
logic of, 173, 186; multiple, 131, 141, 172;
Quine on, 193 ff.; translation of, 141, 176,
187; and variables, 177 ff., 194 S.; and
variables in word language, 191 ff.; in
word language, 186 ff.

Moore, G. E., on analysis, 63
Morris, C. W., 6 n., 305
Multiplicity of entities: in Frege and Church,
i3of., 138; in our method, 17, 145 f., 167 f.

N, 49. 173

% 170

Name, 97 f., 137
Name languages, 75
Name-relation, 97
Names of expressions, 4
NC, Nc, 116

Necessary equivalence, 177
Necessary implication, 177
Necessity, 49, 174
Negation, 3

Neutral expressions, 1S3, iS7

Neutral metalanguage, 153

Neutral variables, 160

Nominalism, 43

Nominatum, 97

Nonextensional expressions, 48

Nonsemantical (absolute) use of semantical

terms, 24, 66; see also Equivalence (2);
L-determinacy (2); L-equivalence (2);
Truth (2)

Null class, 84, 92
Null thing, 36
Number, 43 f., 76, iij, 301
Number concept. Number extension. Num-
ber intension, 117

Object language, 3 f.

Oblique context, 123, 131
Ontology, 22, 43, 196 f., 198
Operand; see Scope
Operators, 3; see also Quantifiers

P, 45
Pt, 28, 94

Paradox of analysis, 63

Paradoxes; see Antinomies
Part-whole relation, 36
Physical magnitude, 85
Platonic ideas, 23
PM, 107, 113, ii4f., 14SS.

PM', 107, 1 13, 148 f., 199
PM", is8

Port-Royal Logic, 126

Positional properties and relations, 75, 80,

84, 92. ttt

Positions, 74
Possibility, 17$

Possible and impossible things, 65
Practical questions, 33, 43, ia8, 136, 168, 190,

304

Predicates, 4
Predicators, 6

Principles; of interchangeability, gif., 177
(see also Frege, Quine, Russell); of the
name-relation, 98; of subject matter, 98;
of univocality, 98
Probability, 90.

Progression, 75
Proper names, 74

Properties, i6ff., 181 f.; nonmental, 20;

physical, 20
Property names, 107

Propositions: false, 28 ff.; as intensions, 27;

nonmental, 37; as ranges, 181
Pseudo-statements, 22, 43
Psychological sentences, 62
Pure modalities, 1 75

Quantifiers, 3

Q 0 inE, W. V.: on abstraction expressions,
103; on antinomy of name-relation, 134,
137. 193; on classes, 104, 1460.; on de-
scriptions, 36, 38; on ‘designation,’ 137;
on ‘designatum,’ 166 n.; on elimination of
antinomies, 161 n.; explicanda, 127; on
identity, 104!.; on individual concepts
197; on individual constants, 74; on indi-
viduals, 103; on meaning, 100, 152; method
of name-relation, 137, 143; on modalities,
193 ff.; on neutrality, 157; on nominalism,
43; on nonextensional contexts, 137;
‘not purely designative,’ 137, 194; on
number, 197; on ontology, 42 f., 1460.,
1961.; on properties, 15a; publications of,
3, 305; on “repudiation” of extensions and
individuals, 197 ff.; rule of context, 147,
150; statement of, 196 f.; on substitutivity,
99, 105; on synonymity, 60, 152; on sys-
tem ML, 102, 141; on values of variables,

7, 42. 196

Quotation marks, 4, 17 n,, 169

RA, 4
Range, 9

Real-number expressions, 79 1 .

Recognition of entities, 22, 43 f., 196 ff.
Reduction of entities, 91 ; of classes to proper-
ties, 146 ff.; of extensions to intensions,

91, isi

aio

IND6X

Relations, 21 , 182

“Repudiation”: of extensions, of-

individuals, 197®.; of irrational numbers,

198 ff.

Rules, semantical, 5, 169; of context, 146 f.,
149, 150; of designation, 4, 5, 176; of
formation, 5, 169 f .; of ranges, 9, 182 ff.;
of truth, 5, 170

RUSSELL, B.; antinomy of, 135; on antino-
my of name-relation, 134, 138ff,;in bibliog-
raphy, 205; on class variables, 113, 20i;on
definition of classes, 107, 147; denial of
meaning, 140, 144; on denoting, 97 n.; on
descriptions, 8, 34 f., 38, 139; duplication of
names, 107 ff.; on extensionality, 1410.;
on Kregc, 118, 140; on identity, 14; on im-
ossible objects, 65; on incomplete sjnn-
ols, 139, 149; on individual constant!!,
74; on name-relation, 139 f.; on number,
8, n5 f., 201; principle of interchange-
ability, 134; on propositional attitudes, 62;
on propositions, 29, 132; rule of context,
149, 150; semantical method of, 138 ff.,
204: on truth-functions, 49

5., 3 ff-

bj, 49. >74. "82

5.1. 75
Sp, 79
S, 4

4

Scope, 32

Semantical methods; see Methods
Semantical rules; sec Rules, semantical
Semantical systems, 3, 169; see also S., etc
Semantics; in cxtensional language, 168 ff.;

in neutral language, 161 ff.

Sentences, 5

Sentential function or matrix; see Matrix
Solutions of antinomy of name-relation,

136 ff-

Specific propositions, 28

Standard forms: of individual e.vpressions,
75, 70, of real-number expressions, 79!
State-descriptions, 9, 182
States, 9

Strict implication; see Lewis
Substitutivlty; see Interchangeability
‘Syncategorematic expressions,’ 7
Synonymity, $6, 60, 133
Synthetic judgments, 12

Tarski, A.; in bibliography, 205; on meta-
language, 44; on semantics, 64 n.; on
truth, s n., 6 n.

Terminology, 6 n., 20 ff.

That-clauses, 27 n., 156
-tor, 6 n.

Translation, 59, from M into M', 158
Truth: (i) semantical, of sentences, 5 f., 93;

(2) nonsemaiitical, of propositions, 94, 132
Truth-conditions, 5
Truth-functions, 49

Truth-values: as extensions, 26; as extra-
linguistic entities, 93 f.; as propositions,

94 f-

Tuhino, a. M., 79 n.

Two, II5

Understanding, 20, 119, 202 f.

Uniqueness condition, 33
Universal class, 84
Universal words, 44
Univcrsals, problem of, 22
Univocality, 98

Value extensions, 45
Value intensions. 45
Values of variables, 42
Variables, 3, 42 ff.; and modalities, 177 ff.,
191 ff., 194 ff.

w, 4

WmTE, M., 63

Whitehead, A. N.: 205 (see also PMj;
on class names, 107; on definition of classes,
147; on nonextensional identity, 109; on
system PM, 107
Wittgenstein, L., 9 f., 205
WooDGER, J. H., 3b n.

.3

: 3 , a, 3

V. 3,

• - 3. 13 !'■

3. s. 13 f-
0. 17s

5 , 176 f.

177

SYMBOLS

A, V, 84
a, 113
». 3- 32
K 3

1 13

o', o", etc., 75

{• -!, 83

[ PRINTtO I
IN u&aJI